![Chapter 1
Convergence and asymptotic approximations
to universal distributions in probability
F. Götze and H. Kösters
The limiting distributions of functionals depending on a large number of independent
random variables of comparable size are often universal, leading to a vast number of
convergence and approximation results. We discuss some general principles that have
emerged in recent years. Examples include classical and entropic central limit theo-
rems in classical and free probability, distributions of zeros of random polynomials
of high degree and related distributions of algebraic numbers, as well as global and
local universality results for spectral distributions of random matrices.1
1.1 Introduction
In random matrix theory, the distributions of the eigenvalues of various kinds of ran-
dom matrices are investigated, see e.g. [3]. More precisely, we are interested in the
asymptotic behaviour of the eigenvalues as the matrix size tends to infinity.
Let us begin with one of the most famous results in random matrix theory, the
semi-circle law. For each n 2 N, let Xn D .n 1=2
Xjk/16j;k6n be a symmetric
Wigner matrix, i.e. a symmetric n n matrix such that .Xjk/16j 6k6n is a family
of independent and identically distributed (i.i.d.) real random variables satisfying the
moment conditions
EXjk D 0 and EX2
jk D 1 (1.1.1)
and, for some ı 0,
sup
n2N
sup
16j;k6n
EjXjkj2Cı
1 : (1.1.2)
The random variables Xjk may depend on n, but this dependence is suppressed in the
notation. Let us mention that all the results described in this section continue to hold
for symmetric random matrices with non-identically distributed entries, and also for
Hermitian Wigner matrices.
Let 1; : : : ; n denote the eigenvalues of Xn, and write Xn
WD 1
n
Pn
j D1 ıj
for
the (empirical) spectral distribution of Xn. The investigation of the limiting spectral
1Projects A4, B1](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-20-320.jpg)
![Convergence and Asymptotic Approximations 3
by deleting the jth row and the jth column, and En is an error term (see below). If
this error term were zero, (1.1.6) would yield the self-consistency equation
m2
.z/ C zm.z/ C 1 D 0 ; (1.1.7)
whose solution is given by the Stieltjes transform of the semi-circle distribution.
Hence, to prove the semi-circle law, one must show that En ! 0 as n ! 1 (in
the appropriate sense). More precisely, it is not hard to see that the error term in
(1.1.6) is given by
.z C mXn
.z//En D 1
n
n
X
j D1
Rjj
n 1=2
Xjj
1
n
X
k;l¤j Wk¤l
XjkXjlR
.j /
kl
1
n
X
k¤j
.X2
jk 1/R
.j /
kk
C 1
n
tr.Rn R.j /
n /
;
(1.1.8)
so it remains to show that the sums on the right-hand side in (1.1.8) tend to zero
as n ! 1. To this end, results from classical probability theory concerning the
moments and distributions of (possibly random) linear and quadratic forms of inde-
pendent random variables are useful.
The problem of optimal approximations of distributions of linear and nonlinear
functions of independent random variables is a classical problem of probability and
harmonic analysis, which originated from classical analytic number theory. For linear
functionals, the arithmetic structure of the summands strongly influences the distri-
bution of the sums, as in the Littlewood–Offord problem [96] and in the central limit
theorem (CLT). For nonlinear functionals like quadratic forms, this influence is al-
ready reduced and studied intensively in the value distribution on lattices by Landau
(1924) and the multivariate CLT for balls by Esséen (1945). Hence, one expects
that distributions of generic highly non-linear functionals of vectors and matrices of
independent variables (e.g. eigenvalues of matrices) exhibit a smooth distributional
behaviour irrespective of a possible arithmetic/lattice structure in the distribution of
these variables. This expected behaviour applies as well to the distribution of roots
of high-degree polynomials with independent random coefficients, which is related
to the distribution of algebraic numbers of growing height.
Random matrix theory is also related to free probability [3, 102]. To explain this,
let us start from the observation that, by generalisation of Eq. (1.1.4), the calculation
of the expected traces of more general products of random matrices is also of interest.
If .Xn/ and .Y n/ are independent sequences of Wigner matrices as in (1.1.4), it turns
out that
lim
n!1
E
1
n
tr
X
j1
n . lim
!1
1
E tr X
j1
/In
Y
k1
n . lim
!1
1
E tr Y
k1
/In
X
jm
n . lim
!1
1
E tr X
jm
/In
Y
km
n . lim
!1
1
E tr Y
km
/In
D 0 (1.1.9)
for all m 2 N and all j1; : : :; jm; k1; : : :; km 2 N. By linearity and by induction, this
allows us to reduce the expected traces of the products Xj1
n Y k1
n Xjm
n Y km
n to those](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-22-320.jpg)
![4 F. Götze, H. Kösters
of the powers Xj
n and Y k
n. The relation (1.1.9) is called the asymptotic freeness of
the sequences .Xn/ and .Y n/.
The relation (1.1.9) is the asymptotic counterpart of the notion of freeness in free
probability. Here, one considers a unital algebra A endowed with a tracial unital
linear functional ' W A ! C and studies the moments '.an
/ of elements a 2 A.
Two elements a; b 2 A are called free (with respect to each other) if
' .aj1 '.aj1 //.bk1 '.bk1 // .ajm '.ajm //.bkm '.bkm //
D 0
(1.1.10)
for all m 2 N and all j1; : : :; jm; k1; : : :; km 2 N. Just as above, this allows us to
reduce the mixed moments '.aj1 bk1 ajm bkm / to the moments '.aj
/ and '.bk
/.
The notion of freeness may be regarded as a (non-commutative) analogue of the
notion of independence in classical probability theory. Moreover, it allows for the
development of a free probability with many parallels to classical probability. For
instance, if a and b are free, the moments of the sum aCb and the product ab depend
only on the moments of a and b. Thus, provided that all the moment sequences
correspond to probability measures a, b
etc. on the real line, one may define the
free additive and multiplicative convolution of a and b by setting
a b WD aCb and a b WD ab :
For the multiplicative convolution, we shall assume without further notice that the dis-
tributions a and b
are supported on the positive half-line. One may then investigate
similar questions (limit theorems, infinite divisibility, asymptotic approximations and
expansions, . . . ) as for the classical convolutions. For instance, the free CLT shows
that if '.a/ D 0 and '.a2
/ D 1, then limn!1 n
a=
p
n
D sc weakly, which is the
direct analogue of the classical CLT. Thus, the semi-circle distribution plays a similar
role in free probability as the normal distribution in classical probability.
When the probability measures a and b
have compact support, one may also
take an analytic approach based on suitable transforms, namely Voiculescu’s R- and
S-transforms [3, 102], instead of the combinatorial approach outlined above. These
transforms are analytic on certain domains in the complex plane and satisfy
Rab
D Ra
C Rb
and Sab
D Sa
Sb
;
and hence may be viewed as analogues of the logarithmic Fourier transform and the
Mellin transform in classical probability theory. Incidentally, the R- and S-trans-
forms are also closely related to the Stieltjes transform in (1.1.5). For instance, for
the R-transform, we have Ra
.z/ D . ma
.z//h 1i
z 1
, where . /h 1i
denotes
the inverse function.
In terms of random matrices, the free convolutions may be interpreted as fol-
lows: Suppose that .Xn/ and .Y n/ are sequences of self-adjoint random matrices of
increasing dimension whose mean spectral distributions converge to X and Y in
moments and which are asymptotically free. (Roughly speaking, this means that their](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-23-320.jpg)
![6 F. Götze, H. Kösters
or for the spectral distribution,
n WD kXn
sck1 :
The problem to establish upper bounds on n has a long history. In 1993, Bai [7]
derived the rate O.n 1=4
/ under a 4th moment condition. The rate O.n 1=2
/ was
obtained independently by Girko (1998, 2002), Bai, Miao and Tsay (2002), and Götze
and Tikhomirov (2003) under various moment conditions; see [69] for references.
Next, the optimal rate O.n 1
/ was obtained for the special case of random matrices
with Gaussian entries, first in the Hermitian case [65] and then in the symmetric case
[100]. Using concentration of measure techniques, Bobkov, Götze and Tikhomirov
[22] obtained the rate O.n 2=3
/ under the assumption that the matrix entries satisfy
a Poincaré inequality. The optimal rate of convergence O.n 1
/ under weak moment
conditions was finally established by Götze and Tikhomirov [68, 69], initially under
an 8th moment condition [68] and later under a .4 C ı/th moment condition [69].
More precisely, it was shown in [69] that if
M4Cı WD sup
n2N
sup
16j;k6n
EjXjkj4Cı
1 (1.2.1)
for some ı 0, then there exists a constant C D C.ı; M4Cı / such that
n 6 Cn 1
(1.2.2)
for all n 2 N.
The proof of this result required three major ingredients:
1. A suitable variant of the smoothing inequality to bound the Kolmogorov distance
in terms of the difference of the corresponding Stieltjes transforms, see Proposi-
tion 2.1 in [68]. This inequality uses a special contour which stays away from
the end-points ˙2 of the support of sc, where the Stieltjes transforms are more
difficult to control.
2. A recursive argument to derive good bounds on the diagonal entries of the resol-
vent close to the real axis, see Section 5 in [68]. Roughly speaking, this argument
shows that a bound on EjRjj j2p
at distance v from the real axis entails a bound
on EjRjj jp
at distance v=s0 from the real axis. It was inspired by similar results,
albeit under stronger moment conditions, by Cacciapuoti, Maltsev and Schlein
[26]. The proof under weak moment conditions is more involved, and uses recur-
sive expansions for the resolvent entries similar to (1.1.6), as well as Burkholder’s
inequality for martingale difference sequences to estimate the resulting quadratic
forms.
3. A recursive inequality for EjmXn
.z/ mSC.z/j2
, see Lemma 7.24 in [68]. This
inequality is based on a Stein-type expansion adapted to the self-consistency equa-
tion (1.1.7), which facilitates the recursion argument considerably.
Using moment matching techniques, a simplified proof of (1.2.2) could be given
in [61].](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-25-320.jpg)
![Convergence and Asymptotic Approximations 7
Let us now turn to upper bounds on
n. In 1997, Bai, Miao and Tsay [9] ob-
tained the rate OP.n 1=4
/ under a 4th moment condition. The rate OP.n 1=2
/ was
established in [64] under a 12th moment condition and in [10] under a 6th mo-
ment condition. More recent results by Erdős, Yau and Yin [42] imply that
n D
OP.n 1
.log n/C log log n
/, see e.g. Section 1 in [68]. In [60], complemented by addi-
tional material in [58, 59], Götze, Naumov and Tikhomirov proved that, under the
condition (1.2.1),
n D OP.n 1
log
n/ ; (1.2.3)
with some explicit constant D .ı/. In view of a result by Gustavsson [72] for
GUE matrices (i.e. Hermitian Wigner matrices with Gaussian entries), it seems clear
that the optimal rate cannot be better than OP.n 1
log1=2
n/. Thus, a result of the
form (1.2.3) is optimal up to logarithmic factors.
Götze, Naumov, Tikhomirov and Timushev [61] improved the result (1.2.3) by
showing that it is possible to take D 2. More generally, they showed that, un-
der the condition (1.2.1), there exist positive constants C D C.ı; M4Cı / and c D
c.ı; M4Cı/ such that for 1 6 p 6 c log n, one has
P.
n K/ 6
Cp
log2p
n
Kpnp
for all K 0 :
These results for
n were obtained by refining the methods developed in [68]. For
instance, in [60], the previous estimates from [68] were extended to the off-diagonal
entries of the resolvent, and Stein-type expansions were employed systematically in
order to bound the pth moment of the error term En in (1.1.8). Moreover, in [61],
the authors used moment matching techniques (motivated by results in [37, 83]) to
compare a general Wigner matrix to a suitable Wigner matrix with sub-Gaussian
entries. Finally, an essential ingredient in all these works was an appropriate version
of the local semi-circle law, which will be described in the next subsection.
1.2.2 Local semi-circle law As mentioned below (1.1.5), the proof of the semi-
circle law amounts to showing that jmXn
.u C iv/ msc.u C iv/j ! 0 as n ! 1,
for any fixed u 2 R, v 0. In the last few years, similar results have been obtained
for the situation where v may tend to zero as n tends to infinity, but not too fast
[40, 38, 60, 61]. More precisely, under suitable moment conditions,
jmXn
.u C iv/ msc.u C iv/j 6
.log n/C log log n
nv
(1.2.4)
with high probability, uniformly in u 2 R and 1 v log n=n, say. A result of the
form (1.2.4) is called a local semi-circle law in the literature, since it may be used as
a starting point for the investigation of the local distribution of the eigenvalues.
In fact, the local semi-circle law was originally developed by Erdős, Schlein and
Yau [40] on their way to proving the universality of the local spectral distribution of](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-26-320.jpg)
![8 F. Götze, H. Kösters
general Wigner matrices, see also Section 1.4. The first version of the local semi-
circle law was derived under the assumption that the underlying matrix entries have
finite exponential moments (as well as further regularity properties). This assump-
tion was successively relaxed to .4 C ı/th moments in a series of papers by Erdős,
Knowles, Schlein, Yau and Yin; compare [38] and the references therein.
The paper [60] provided an alternative self-contained proof of the local semi-
circle law, also under a .4 C ı/th moment condition, by building upon the techniques
developed for the investigation of the rate of convergence [68]. One of the main
advantages of this approach is that the exponent of log n in (1.2.4) is reduced from
C log log n to a constant (at least in a certain region for the arguments u and v), with
a precise dependence on ı.
These results were further improved by Götze, Naumov, Tikhomirov and Timu-
shev [61], who showed that for any ı 0, there exist constants C0; C1; C2 depending
only on ı and M4Cı (see (1.2.1)) such that
EjmXn
.u C iv/ msc.u C iv/jp
6
C0p
nv
p
for all 1 6 p 6 C1 log n, 1 v C2 log n=n and juj 6 2 C v. By taking
p of the order log n and using Markov’s inequality, one re-obtains (1.2.4) for 1
v C2 log n=n and juj 6 2 C v, but with the constant exponent D 1 instead of
D C log log n for the logarithmic factor.
As already mentioned, this version of the local semi-circle law plays a major role
in recent advances on the rate of convergence for
n. Further applications include
the delocalisation of eigenvectors and the rigidity of eigenvalues; see [42, 60, 61] and
the references given there. In view of the results by Gustavsson [72], all these results
seem to be optimal up to logarithmic factors.
1.3 Non-symmetric random matrices
The spectral distributions of non-symmetric random matrices have also been investi-
gated.
1.3.1 Circular law For each n 1, let Xn D .n 1=2
Xjk/16j;k6n be a real Girko–
Ginibre matrix, i.e. an n n matrix such that .Xjk/16j;k6n is a family of i.i.d. real
random variables satisfying the moment conditions
EXjk D 0; EX2
jk D 1 .1 6 j; k 6 n/
and, for some ı 0,
sup
n2N
sup
16j;k6n
EjXjkj2Cı
1 :
Again, the random variables Xjk are allowed to depend on n. Also, all the results
described in this section continue to hold for non-symmetric random matrices with](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-27-320.jpg)
![Convergence and Asymptotic Approximations 9
non-identically distributed matrix entries, as well as for complex Girko–Ginibre ma-
trices.
Similarly as above, let 1; : : :; n denote the eigenvalues of Xn, and write Xn
for the (empirical) spectral distribution of Xn. Of course, this is in general a proba-
bility measure on the complex plane now.
The famous circular law states that
lim
n!1
EXn
D circ.dz/ WD 1
1
1
1fjzj1gdz .n ! 1/
in the sense of weak convergence, or a similar result without the expectation (with
weak convergence in probability or almost surely). Thus, the spectral distribution
converges to the uniform distribution on the unit disk in the complex plane, and the
limiting distribution is again universal.
This result was first stated by Girko in the 1980’s. One of his key ideas was
to study the spectral distributions of the matrices Xn by considering the spectral
distributions of the Hermitian random matrices .Xn zIn/.Xn zIn/
, for all
z 2 C. More precisely, for a probability measure with compact support on the
complex plane, let
U.z/ WD
Z
C
log j zj.d/ .z 2 C/ (1.3.1)
denote the associated logarithmic potential. We shall write UXn instead of UXn
.
Then, it is easy to see that
UXn
.z/ D
Z 1
0
log.t/.Xn zIn/.Xn zIn/ .dt/ .z 2 C/ ; (1.3.2)
and the proof of the circular law boils down to proving the convergence of the corre-
sponding logarithmic potentials.
It is worth emphasising that the logarithmic potential UXn .z/ is given by an inte-
gral over an unbounded function. Thus, to prove the convergence of (1.3.2), say, it is
not sufficient to establish the weak convergence of the singular value distributions of
the matrices X zIn, but one must additionally control the large and small singular
values of these matrices. The real problem is that of controlling the small singular
values. In the case where the matrix entries have a smooth density, this problem was
solved by Bai [8]. For general distributions, however, this problem remained open. Its
solution became possible only recently due to work by Rudelson [91] and Rudelson
and Vershynin [92], who derived stochastic lower bounds on the smallest singular
value of a non-symmetric random matrix with independent entries. These bounds
were obtained by a combination of geometric and probabilistic methods, inter alia
concentration inequalities for linear forms of independent random variables. Inde-
pendently of each other, Götze and Tikhomirov [66] and Tao and Vu [97] extended
these bounds to ‘shifted’ matrices Xn zIn and applied them to complete the proof
of the circular law.](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-28-320.jpg)
![10 F. Götze, H. Kösters
To continue, optimal bounds for the concentration in small balls of weighted sums
Pn
kD1 Xk
ak
of vectors ak
with i.i.d. real weights Xk
crucially depend on the arith-
metic properties of the vectors ak if Xk are Rademacher variables taking values ˙1.
Arithmetic consequences of strong concentration have been investigated by Nguyen,
Tao and Vu (2009, 2011) e.g. in [96, 87]. They called this inverse Littlewood–Offord
phenomenon.
Eliseeva, Götze and Zaitsev [35, 36] showed that this phenomenon can be de-
rived from seminal results of Arak (1980, 1981) and Arak and Zaitsev (1988), see [4]
and the references therein, who investigated a similar phenomenon in a more general
context. They studied how a large local concentration of the sum Sa WD
Pn
kD1 Xk
ak
of i.i.d. variables Xk implies a good approximation of the distribution of Sa by an
induced measure of distributions supported on a higher dimensional lattice under a
linear map. As shown in [35, 36], this yields information about the arithmetic struc-
ture of the sequence ak extending the results of [96, 87].
1.3.2 Elliptic law After the proof of the circular law, various extensions were con-
sidered, see also the survey paper by Bordenave and Chafaï [23].
One line of research was the proof of the elliptic law [85, 86, 55]. Here, one con-
siders random matrices Xn D .n 1=2
Xjk/16j;k6n with similar moment properties
as in the circular law, but for which the random variables Xjk and Xkj with j ¤ k
are correlated, with a fixed correlation coefficient % 2 . 1; C1/. It turns out that
the asymptotic spectral distribution is given by the uniform distribution on an ellipse,
with a shape determined by %. By and large, the proof of the elliptic law resembles
that of the circular law. However, due to correlations, several parts of the proof have
to be adapted, including the bounds on the small singular values.
Note that one may regard the ‘elliptic’ matrices as interpolations between Wigner
matrices and Girko–Ginibre matrices, and their limiting distributions as interpolations
between the semi-circular distribution and the circular distribution.
1.3.3 Products of random matrices In a different direction, the spectral distribu-
tions of products of independent Girko–Ginibre matrices X1;n Xk;n (all of size
n n) were investigated, for k fixed and n ! 1. Similarly as for Girko–Ginibre
matrices, both the singular value distributions [2] and the eigenvalue distributions
[67, 90] were studied.
The distribution of the (squared) singular values may be addressed either via the
method of moments, which leads to the Fuss–Catalan numbers, or via the method
of Stieltjes transforms, which leads to an algebraic equation of degree k C 1 for the
Stieltjes transform of the limiting distribution.
After that, the derivation of the eigenvalue distribution again uses Girko’s Hermiti-
sation method. Here, the small singular values of the ‘shifted’ matrices X1;n Xk;n
zIn may be controlled by using the previous results for a single Girko–Ginibre ma-
trix Xn as well as Horn’s inequalities for the singular values of product matrices, see
e.g. [73, Thm. 3.3.14]. The final result is that, for fixed k, the limiting eigenvalue](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-29-320.jpg)
![Convergence and Asymptotic Approximations 11
distribution is universal and given by
.k/
circ.z/ D
1
kjzj2.k 1/=k
1
1
1fjzj1gdz : (1.3.3)
Remarkably, this is the same limiting distribution as that for the kth power of a single
Girko–Ginibre matrix.
Furthermore, the same result holds for products of independent ‘elliptic’ random
matrices [56, 89]. Somewhat surprisingly, for products with k 2 factors, one also
obtains the limiting eigenvalue density (1.3.3), irrespective of the correlation % of the
underlying factors.
Fluctuation results for linear statistics of singular values of products of random
matrices have also been studied [57].
1.3.4 Functions of random matrices Götze, Kösters and Tikhomirov [53] pro-
posed a general framework for the investigation of matrix-valued functions
F n.X1;n; : : :; Xk;n/ of independent Girko-Ginibre matrices X1;n; : : : ; Xk;n (all of
size n n). One of the main ideas was to exploit the principle of universality, thereby
partly avoiding the need to analyse the self-consistency equations for the Stieltjes
transforms.
In a first step, one identifies a set of conditions under which one may establish
the universality of the limiting singular value and eigenvalue distributions. The in-
vestigation of the singular values relies on the Stieltjes transform and uses classical
truncation and interpolation arguments from probability theory, while the study of the
eigenvalues is again based on Girko’s Hermitisation method.
In a second step, one determines the limiting spectral distributions in a simple
special case, namely for functions of Gaussian random matrices. Here one can take
advantage of the fact that these matrices are bi-orthogonally invariant (i.e. their dis-
tribution is not changed if we pre- or postmultiply them by orthogonal matrices), and
hence asymptotically free. Thus, it becomes possible to determine the limiting spec-
tral distributions by using the calculus of S- and R-transforms from free probability
(see the Introduction) as well as classical results from logarithmic potential theory, at
least in simple situations.
It turns out that the underlying assumptions may be verified in a number of exam-
ples. In particular, many results on products of independent Girko-Ginibre matrices
and their inverses may now be derived within a common framework [53]. Further-
more, the same approach may also be used for sums of independent random matrices
[99, 79], although a more explicit description of the limiting spectral distributions is
available only in a few special cases here. For example, for random matrices of the
form X1;n.X2;n Xk;n/ 1
, or even sums of independent copies thereof, the limit-
ing spectral distributions are closely related to the symmetric stable distributions from
free probability [79].](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-30-320.jpg)
![12 F. Götze, H. Kösters
1.4 Local spectral distributions
Let us turn to the local spectral distributions. For later comparison, we start with the
classical results for the Gaussian unitary ensemble (GUE), see e.g. [3]. This is the
probability distribution on the space Hn of Hermitian n n matrices with the matrix
density
const exp 1
2
n tr X2
dX ; (1.4.1)
where dX D
Q
16j 6k6n dX
jk
Q
16j k6n dX=
jk
denotes Lebesgue measure on Hn.
GUE matrices are special in that they are unitarily invariant (i.e. their distribution
does not change under conjugation by unitary matrices) and of Wigner type (i.e. their
entries are independent random variables up to the Hermiticity constraint). Further-
more, the joint distribution of the eigenvalues may be calculated explicitly. It is given
by the density
const
Y
16j k6n
jxk xj j2
n
Y
j D1
exp 1
2
nx2
j
dx1 : : : dxn ; (1.4.2)
which may be rewritten in determinantal form,
det Kn.xj ; xk/
j;kD1;:::;n
dx1 : : :dxn : (1.4.3)
Here, Kn is a certain kernel given by a sum of products of Hermite polynomials.
Due to this representation, the asymptotic behaviour of the spectral distribution of a
GUE matrix Xn follows from that of the Hermite polynomials, which is well known.
For the global spectral distribution of the matrices Xn, one obtains (of course) the
density of the semi-circular distribution, %sc.x/ D 1
2
p
.4 x2/C. For the local
spectral distribution, one must first rescale the eigenvalues so that the mean spacing
between neighbouring eigenvalues is of the order 1. In the bulk of the spectrum, i.e.
near a point a 2 . 2; C2/ with %sc.a/ 0, the properly rescaled eigenvalues are
given by Q
j .Xn/ D n%sc.a/.j .Xn/ a/. One then finds that the local correlations
are asymptotically given by the determinant of the sine kernel
Ksine.x; y/ WD
sin .x y/
.x y/
:
That is, for any m 2 N and any smooth function f W Rm
! R with compact support,
we have
lim
n!1
E
0
@.n m/Š
nŠ
X
j1;:::;jm
f .Q
j1
.Xn/; : : :; Q
jm
.Xn//
1
A
D
Z
Rm
f .x1; : : : ; xm/ det Ksine.xj ; xk/
j;kD1;:::;m
dx1 dxm ;
(1.4.4)](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-31-320.jpg)
![Convergence and Asymptotic Approximations 13
where the sum inside the expectation is over all m-tuples with pairwise distinct el-
ements. Similar results, although with a different rescaling and a different limiting
kernel, the Airy kernel, hold at the edge of the spectrum, i.e. near the points a D ˙2.
Remarkably, the asymptotic local distributions are universal in the sense that the same
limits arise in a variety of other situations, both inside and outside the field of random
matrix theory.
1.4.1 Random matrices The same local correlations have been established for
several classes of random matrices. First of all, it has been known for some time
now that similar results hold if we replace the GUE with a unitarily invariant random
matrix ensemble defined by a potential V with suitable smoothness and growth
properties. Here the matrix density and the eigenvalue density are given by
const exp. n tr V.X// dX and
const
Y
16j k6n
jxk xj j2
n
Y
j D1
exp. nV.xj //dx1 : : : dxn ; (1.4.5)
respectively. Note that in the special case V.x/ D 1
2
x2
, we re-obtain the results
for the GUE. A density of the form (1.4.5) is also called an orthogonal polynomial
ensemble. The reason is that one can use the orthogonal polynomials associated with
the weight e nV.x/
to rewrite the density (1.4.5) in determinantal form, with a kernel
Kn;V , similarly as in (1.4.3). Thus, the asymptotic analysis is again reduced to ortho-
gonal polynomials, which can now be analysed e.g. using Riemann–Hilbert problems,
see e.g. Deift [34] and the references therein. At the global level, one obtains a
deterministic limit density %V with compact support. It is worth emphasising that this
density is not universal, but depends on the choice of the potential V . At the local
level, after replacing %sc with %V in the rescaling of the eigenvalues, one obtains the
same result (1.4.4) as for the GUE in the bulk of the spectrum. Moreover, similar
universality results (with the sine kernel replaced by the Airy kernel) hold at the edge
of the spectrum. Thus, the local correlations are universal within the class of unitarily
invariant ensembles. Strong universality results for the kernel Kn;V including rates
of convergence and transitions between bulk and edge regions have been obtained in
[80].
For more general random matrix ensembles, however, a closed expression for the
joint distribution of the eigenvalues as in (1.4.2) or (1.4.5) is not available anymore,
and the analysis becomes considerably more complicated.
Johansson [74] investigated the local correlations for the so-called deformed
Gaussian unitary ensemble and proved the analogue of (1.4.4). A deformed GUE
matrix has the form Xn C aY n, with Xn a Hermitian Wigner matrix, Y n an inde-
pendent GUE matrix, and a 0 fixed. The investigation of this ensemble starts from
two important observations. Firstly, the eigenvalue density may be represented as a
mixture of certain determinantal densities. Secondly, the correlation kernels associ-
ated with these determinantal densities admit a double contour integral representation
which is suitable for asymptotic analysis.](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-32-320.jpg)
![14 F. Götze, H. Kösters
Götze, Gordin and Levina [47, 46] investigated the local correlations for a partic-
ular fixed-trace ensemble, the so-called fixed Hilbert–Schmidt norm ensemble (HSE),
and proved the analogue of (1.4.4). For r 0, this ensemble is given by the unique
probability measure on the set fX 2 Hn W tr.X2
/ D nr2
g which is invariant with
respect to conjugation by unitary matrices. The main idea is that the GUE may be
represented as a mixture of the HSE, for different values of r, and that this relation
may be inverted in order to deduce the results for the HSE from the corresponding re-
sults for the GUE. The implementation of this idea is technically involved, however,
and requires extending the parameter r to the complex domain.
More recently, it has been proved by Erdős, Péché, Ramírez, Schlein and Yau [39]
and by Tao and Vu [98] that (1.4.4) continues to hold for general Hermitian Wigner
matrices satisfying certain technical conditions. Consequently, the local correlations
are universal within the class of Hermitian Wigner ensembles as well. The proof of
this result is based on sophisticated comparison arguments. The main idea in [39] is to
compare general Wigner matrices to random matrices from a suitable deformed GUE,
but with a ! 0 as n ! 1, and to show that the local correlations are asymptotically
the same. To this end, one needs the local semi-circle law (and several related results)
as an input. We refer to the survey paper [41] for details.
1.4.2 Products of random matrices It is well known that the singular value and
eigenvalue configurations of a Ginibre matrix (i.e. a non-Hermitian random matrix
with i.i.d. complex Gaussian entries) have spectral densities of determinantal form.
More recently, it has been observed by Akemann, Burda, Ipsen, Kieburg, Wei and
others that the same is true for products of independent Ginibre matrices; compare [1]
and the references therein. Up to rescaling, the joint density of the squared singular
values of a product of m independent Ginibre matrices is given by
const det.xk 1
j /j;kD1;:::;n det..Dj 1
w/.xk//j;kD1;:::;n ; (1.4.6)
where .Dw/.x/ WD xw0
.x/ and w denotes a certain weight function. For m ¤ 1,
this is not an orthogonal polynomial ensemble as in (1.4.5) anymore, but it is still a
so-called bi-orthogonal ensemble [24, 82, 1]. Thus, it is also possible to rewrite the
density in determinantal form and to investigate the local correlations for m fixed and
n ! 1, see e.g. [1] for an overview. In the bulk and at the edge of the spectrum,
one still obtains the familiar sine and Airy kernel, while at the origin, one obtains a
new kernel which generalises the Bessel kernel known from the case m D 1. For
the eigenvalues, analogous results can be derived. Moreover, similar results were
obtained for product matrices consisting of independent Ginibre matrices, truncated
unitary matrices, their inverses, or combinations thereof.
The original derivations of these results were often based on ad hoc methods. In-
spired by related investigations in [82], Kieburg and Kösters [76, 77] observed that
several of these results can be obtained in a unifying framework by means of the
spherical transform. This is a multivariate analogue of the Mellin transform which is
suitable for the investigation of products of independent bi-unitarily invariant random
matrices. Moreover, the spherical transform also leads to a closer description of the](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-33-320.jpg)
![Convergence and Asymptotic Approximations 15
relation between the squared singular values and the eigenvalues for these matrices
[76]. In particular, it turns out that the class of bi-unitarily invariant random matri-
ces corresponding to a density of the form (1.4.6) is closed under taking indepen-
dent products [77]. This leads to a fairly large class which includes many prominent
ensembles from non-Hermitian random matrix theory and which gives rise to new
universality problems.
1.4.3 Repulsive particle systems Götze and Venker [70] considered more general
probability measures on Rn
with a density of the form
f .x1; : : :; xn/ D const
Y
16j 6k6n
h.xj xk/
Y
16j 6n
e nQ.xj /
dx1 dxn ; (1.4.7)
where h is a non-negative function satisfying h.x/ D x2
.1 C o.1// as x ! 0 and Q
is a potential with sufficient growth at infinity. Note that, in the special case where
h.x/ D x2
, the density f reduces to that of the eigenvalues of a unitarily invariant
random matrix, see (1.4.5). For a general function h, however, the density f does
not seem to have a natural spectral interpretation anymore. Moreover, the correlation
functions are not known to be determinantal. Still, the density may be viewed as
describing an interacting particle system, with a repulsive interaction generalising
that between eigenvalues of (Hermitian) random matrices.
Götze and Venker [70] proved that, under certain technical conditions, the asymp-
totic local correlations of the particles are still given by (1.4.4). The proof is based
on a comparison between the ensemble Pn;Q;h defined by (1.4.7) and an appropriate
eigenvalue ensemble Pn;V as in (1.4.5). In a first step, one uses a fixed point argu-
ment to construct a potential V (depending not only on Q but also on h) such that
Pn;Q;h and Pn;V have the same limiting distribution at the global level. In the sec-
ond step, one shows that the corresponding local correlations are asymptotically the
same. Here, one uses the theory of Gaussian processes to express the change of mea-
sures from Pn;V to Pn;Q;h in terms of linear eigenvalue statistics, which may then
be controlled using concentration of measure inequalities for the probability measure
Pn;V .
Kriecherbauer and Venker [81] derived similar universality results at the edge of
the spectrum, where one encounters the Airy kernel. Universality for the empirical
spacing distribution was shown by Schubert and Venker in [93]. There, also a new lo-
cal rescaling of eigenvalues using distribution functions was introduced, which yields
better rates of convergence than the previously used rescalings. Moreover, Venker
[101] proved a similar result as in [70] for the case where h.x/ D jxjˇ
.1 C o.1// as
x ! 0, with general ˇ 0. In this case, the asymptotic local correlations in the bulk
coincide with those of the Gaussian beta ensemble, which is obtained by replacing
Q
16j k6n jxk xj j2
with
Q
16j k6n jxk xj jˇ
in (1.4.2).](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-34-320.jpg)
![16 F. Götze, H. Kösters
1.5 Connections between probability theory and number theory
1.5.1 Correlations of zeros of random polynomials It is a long-standing conjec-
ture, supported by partial theoretical results [84] as well as by extensive numerical
evidence [88], that the local correlations of the zeros of the Riemann zeta function
along the critical line are also governed by the sine kernel. Moreover, further similar-
ities emerge if one compares the moments and the (auto)correlations of the Riemann
zeta function with those of characteristic polynomials of various classes of unitary
and Hermitian random matrices (Keating–Snaith [75], Brézin–Hikami [25], Conrey–
Farmer–Keating–Rubinstein–Snaith [33], Strahov–Fyodorov [95], Götze–Kösters
[52], Shcherbina [94]).
Connections to number theory also arise in another direction. Given that the
eigenvalues of a random matrix are the zeros of the corresponding characteristic
polynomial, it seems natural to ask for the distribution of the zeros for other classes
of (random or non-random) polynomials as well. See for instance Götze, Kaliada
and Zaporozhets [49] for the distribution of complex algebraic numbers of increasing
height.
Note that the transformation from the coefficients to the zeros of a polynomial
is a highly non-linear operation, just as the transformation from the entries to the
eigenvalues of a matrix. Even more, the Jacobian of both transformations is closely
related to the squared Vandermonde determinant,
Q
16j k6n jzk zj j2
. In view of
this observation, the numerous similarities between zeros of random polynomials and
eigenvalues of random matrices come perhaps not too unexpected.
Recently, Götze, Kaliada and Zaporozhets [50] investigated the correlations of
zeros of random polynomials of degree n of with real coefficients with a joint density.
Since a real polynomial can have both real and complex zeros, with the complex zeros
symmetric with respect to the real axis, it is convenient to consider, for any k; l 2 N0
with k C 2l 6 n, the mixed correlation functions %k;l describing the correlations
between k eigenvalues on the real line and l eigenvalues in the upper half-plane.
These correlation functions also contain a Vandermonde factor, which shows that
adjacent zeros tend to repel one another.
In a different direction, Götze, Kaliada and Zaporozhets [51] investigated the cor-
relations between conjugate algebraic numbers. More precisely, consider all prime
polynomials with integer coefficients of degree n and height bounded by Q, for var-
ious choices of height, and count the corresponding .k C l/-tuples of conjugate al-
gebraic numbers with values in a given subset of Rk
Cl
C. Then, it can be shown
that, after appropriate normalisation, these numbers converge to a limit as Q ! 1.
Furthermore, at least for some choice of the height function, the limiting correla-
tion functions coincide with the known correlation functions of the zeros of certain
random polynomials.
1.5.2 Diophantine approximations in metric number theory The problem of Dio-
phantine inequalities for linear forms on Zd
with coefficients taken from a submani-
fold of Rd
has been studied by Mahler, Khinchine and more recently by Schmidt and
Sprindžuk. In the last decade Margulis, Kleinbock, Bernik and Beresnevich solved](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-35-320.jpg)
![Convergence and Asymptotic Approximations 17
important problems concerning Diophantine inequalities for one-dimensional sub-
manifolds given by .x; x2
; : : : ; xd
/ 2 Rd
; x 2 R, and more generally for higher-
dimensional submanifolds defined by a non-degenerate system of functions. In par-
ticular, the breakthrough solution of the Baker–Sprindžuk conjecture due to Klein-
bock and Margulis [78] that used the dynamics of flows on homogeneous spaces led
to a surge of activity in a range of related long-standing problems. Some advances of
more recent years include progress on the distribution of rational and algebraic points
near manifolds [12, 15, 48], the distribution of discriminants and resultants of integral
polynomials [16] as well as sharp quantitative bounds for the number of pairs of close
conjugate algebraic numbers of a fixed degree n and bounded height [13].
More precisely, Beresnevich, Bernik and Götze [13] studied for arbitrary con-
jugate algebraic numbers over Q (resp. algebraic integers) ˛1 ¤ ˛2 the universal
minimal exponent, say D n (resp. D
n ), such that the inequality j˛1 ˛2j
H.˛1/
holds for sufficiently large H.˛1/. Here H.˛/ denotes the height of the
algebraic number ˛, i.e. the absolute height of the minimal polynomial of ˛ over Z.
Clearly
n 6 n. Mahler proved in 1964 that n 6 n 1. For n 6 3, satisfac-
tory bounds have been shown by Bugeaud, Evertse and Mignotte. An old result by
Mignotte for the general case, i.e., n 3 said that n;
n n=4. In [13] it was shown
that in fact, for n 2, n;
n .n C 1/=3 holds, which follows from a counting re-
sult for pairs of real algebraic conjugate numbers in certain intervals of given large
height. This is in turn a consequence of results in metric number theory which gen-
eralises results of Baker, Schmidt, Bernik, Kleinbock and Margulis. For an overview
of techniques and results, see [14].
1.5.3 Effective bounds in Diophantine inequalities A classical problem of effec-
tive bounds means to determine the size of the smallest vector m 2 Zd
n f0g such
that the Diophantine inequality jQŒmj 1 holds. Here, QŒx denotes an (non-
degenerate) indefinite quadratic form with real coefficients on Rd
. For forms with
integer coefficients it is known by a result of Meyer (1884) that there exists an
m 2 Zd
n f0g with QŒm D 0 provided that d 5. As for the size of this solu-
tion vector, classical results of Birch and Davenport (1958) using the geometry of
numbers show that for indefinite integral quadratic forms 0 QCŒm cd j det Qj
holds, where QC D .QT
Q/1=2
is positive definite.
For diagonal indefinite quadratic forms Q, Birch and Davenport (1958) have
shown that there exists a nontrivial solution m ¤ 0 to jQŒmj of size kmk ı
2Cı
in dimension d D 5. In higher dimensions solutions may be generated via
embedding. Using theta functions, methods from the geometry of numbers and
the asymptotic orbit behaviour of unipotent subgroups of SL2.R/, Götze and Mar-
gulis proved in [54] for general indefinite QŒx a bound of order 0 kmk ı
kd Cı
; kd D 12; 8:5; 7 for d D 5; 6 and d 7.
1.5.4 Lattice point counting problems The existence of m ¤ 0 with jQŒmj is
a consequence of precise results for counting lattice pointsin elliptic as well as hyper-
bolic shells defined via a quadratic form QŒx in Rd
and Ea;r WD fQŒx 2 Œa; rg for](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-36-320.jpg)
![18 F. Götze, H. Kösters
Q positive definite and Hr .a; b/ WD fQŒx 2 Œa; b; x 2 rCg (C suitable compact
convex body) in the case of Q indefinite as r in Hr .a; b/ and Ea;r tends to infinity.
The counting error is measured relative to volume, say jE0;rj, resp. jHr .a; b/j of
these compact rescaled regions, say Br , via
ˇ
ˇ card.Zd
Br / jBr j
ˇ
ˇ = jBrj D O.ır r 2˛
/; r ! 1; (1.5.1)
where ˛ 6 1. In case that the optimal exponent ˛ D 1 can be shown, the factor ır
may be bounded below or tend to zero depending on the Diophantine properties of
the coefficients of Q. Such error bounds have been studied for a long time, starting
with Landau (1915, 1924), Jarnik (1928) for special positive definite forms QŒx
being diagonal or with integer coefficients, where ˛ D 1 could be shown. Optimal
exponents have been shown for d 9 and definite as well as indefinite general forms
QŒx first by Bentkus and Götze (1997). For dimension d 5, the optimal exponent
was shown in [45] for E0;r and in [54] for Hr .a; b/ such that b a tends to infinity.
As for the dependence of Diophantine properties of Q, Davenport and Lewis (1972)
conjectured that for positive definite irrational forms the distances of the ordered
elements of QŒZd
for d 5 converge to 0. This is related to the famous conjecture
by Oppenheim (1929) that QŒZd
is dense in R for indefinite irrational forms for
d 5, which was proved by Margulis (1986) even for d 3. Quantitative versions
of these conjectures for irrational forms were shown for d 9 in Bentkus and Götze
(1997) as well as for d 5 and Q positive-definite in [45]. In [54], the problem for
both cases has been solved for d 5 by means of a unified approach.
Lower bounds for the error for E0;r in dimensions d D 2; 3; 4 show that the
optimal exponents satisfy ˛ 1=4; 1=2; 1 respectively, multiplied by logarithmic
factors like .log r/ˇ
; ˇ 0; d D 2; 3 resp. .log log r/ ; 0; d D 4, whereas
lower error bounds for dimensions 5 and higher are just given by .r 2
/.
Crucial for the proof of ˛ D 1 and ır ! 0 in (1.5.1) is the investigation of theta
sums on the generalised Siegel upper half-plane, that is on matrices iQ C A, where
A; Q are real symmetric d d matrices with Q positive definite, given by
.iQ C A/ D
X
m2Zd
expΠmT
Q m C imT
A m: (1.5.2)
The number of lattice points m such that Q.m/ 6 r may be expressed as an integral
along a theta function .sQ/ expŒitr=.2s/ on a line s D t C ir 2
; t 2 R (that is
a degenerate horocycle), with given by (1.5.2). The following inequality between
theta functions turned out to be an essential step:
j.sQ/j2
6 .t/ WD c.Q/rd
X
m;n
expΠHt .m; n/; (1.5.3)
where Ht denotes the t-dependent quadratic form
Ht .m; n/ WD r2
km t Q nk2
C r 2
knk2
(1.5.4)](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-37-320.jpg)
![20 F. Götze, H. Kösters
lattice point problems and the CLT and the extensive literature in this field, see the
reviews in [44] and [71].
The approximation results for Hr .a; b/ described above were essential as well
for proving the above rate of convergence ˛ D 1 for quadrics in the CLT down to
dimension 5 in [71]. Here, Götze and Zaitsev proved, for example, an explicit error
bound for the elliptic or hyperbolic regions of type B WD fQŒx 6 1g in dimensions
d 5,
sup
r0
j PfQŒSn 6 nr2
g PfQŒS 6 r2
g j 6 cd n 1
j det Qj 1=2
EkQ 1=2
Xk4
:
This result concludes a long series of investigations starting with the seminal results
by Esséen (1945) mentioned above. It relies on the fact that the characteristic func-
tion of quadratic forms Q.Sn/ for sums of independent identically distributed vector
summands may be estimated via the average over random matrices A in the char-
acteristic functions of bilinear forms hAT1; T2i of two independent sums T1; T2 of
Rademacher vectors taking values in the lattice Zd
. These characteristic functions
in turn may be estimated again using local limit theorems via theta sums as outlined
in (1.5.3). Hence, the decisive step of controlling the local fluctuations of the distri-
bution of such indefinite quadratic forms could be reduced to the estimates outlined
above in order to obtain a full correspondence in terms of dimensions and rates be-
tween both areas. In order to investigate distribution functions of quadratic forms, a
crucial technical obstacle had been to extend the averaging (in t) of the characteristic
described in (1.5.5) from the measure dt (sufficient for narrow shells Hr .a; b/ with
bounded intervals Œa; b) to the harmonic measure dt=t on the real line.
For lower dimensions, for example d D 3, one cannot expect these optimal rates
in view of the correspondence to the lattice point error problem in these dimensions.
1.6 Analogies between classical and free probability
In non-commutative probability theory the addition and multiplication of ‘free ran-
dom variables’ corresponds to the ‘free’ additive and multiplicative convolution of
the corresponding spectral probability measures which are represented by Hermitian
operators in certain von Neumann algebras A with a finite normalised faithful trace
, see e.g. the survey [102]. A guiding principle for the investigation of free convo-
lutions of spectral measures is the analogy to the classical theory of convolutions of
probability measures.
Within the framework of comparison studies between classical and free probabil-
ity, Chistyakov and Götze investigated the free additive convolution with respect to
a classification of indecomposable elements and infinite divisibility. The final results
now appeared in [29]. They represent an analog of the classical results of infinite di-
visibility via the so-called Bercovici–Pata bijection, which yields a correspondence of
classical Lévy measures to free Lévy measures. The latter appear in free Khintchine
type decompositions via integral representations of reciprocal Stieltjes transforms,](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-39-320.jpg)
![Convergence and Asymptotic Approximations 21
which are analytic functions from CC
to C and hence closely related to the class of
Nevanlinna functions. A remarkable difference described for free measure decompo-
sitions in [29] as compared to classical ones is that the infinitely divisible measures
without indecomposable factors are trivial Dirac measures only. This observation
holds not only for the free additive convolution, but also for the free multiplicative
convolution on the positive half-line as well as on the unit circle [29].
As for analogs of results of classical parametric statistics, the independence of
sample mean and sample covariance for Gaussian random variables has a counterpart
in free probability for free semi-circular random variables in von Neumann algebras;
see Chistyakov, Götze and Lehner [32].
1.6.1 Asymptotic approximations in free probability In [28], Chistyakov and
Götze investigated the rate of convergence of the distribution function of an n-fold
normalised free additive convolution of a spectral measure with itself to that of
Wigner’s semi-circle law. Since their result showed a complete analogy to the Berry–
Esséen theorem in classical probability, i.e. a rate of order O.n 1=2
/ assuming the
existence of a third moment of , they investigated higher order approximations for
this n-fold free convolution of non-trivial measures. Unlike the classical case there
are no arithmetic obstructions to the smoothness of such convolutions: for n larger
than a finite threshold (depending on ), the resulting measure admits a density with
respect to Lebesgue measure. Thus one would hope that, similarly to Edgeworth ex-
pansions in classical probability, moment conditions of order k C 2 suffice to define
asymptotic approximations up to an error of order o.n k=2
/ involving free cumulants.
This is indeed the case for k D 1, and the expansion term involves derivatives of the
semi-circle density. Differences to the classical case appear for the expansion term
of order k D 2, which cannot be written in terms of derivatives of the semi-circle
density anymore: one needs signed measures and a slightly shifted support when the
third cumulant does not vanish. These problems are essentially caused by the singu-
larity of the semi-circle density at the boundary of the support. Technically the proofs
for the expansions are analytically much more involved then in the classical case. For
k D 1; 2 one has to determine a subordinating function as a particular solution of a
3rd resp. 5th order equation, which degenerates into the quadratic equation (1.1.7) for
the Stieltjes transform of Wigner’s semi-circle law as n ! 1, and study its explicit
dependence on n.
An alternative way to derive expansions of n 1=2
.X1 C : : : C Xn/ in the free
CLT for free identically distributed random variables Xj and related approximation
problems for random matrices starts from a universal scheme of expansions for se-
quences of symmetric functions, see [62]. This scheme is an umbrella limit theorem
for all Gaussian limits and related Edgeworth-type expansions in permutation sym-
metric functions of many variables by rotational invariant functions and corrections
via polynomials in power sums of degree at least two. Applications to the highly
non-linear free convolution were done in [63]. It involved derivatives with respect to
j at zero of distribution functions like W C 1X1 C 2X2, where W has semicircle
distribution. In the interior of the limiting spectral support the validity of these ex-](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-40-320.jpg)
![22 F. Götze, H. Kösters
pansions is due to the smoothness of the distribution functions of 1X1 C : : : C nXn
of .1; : : : ; n/ 2 Rn
in certain domains.
1.6.2 Entropic limit theorems Let us return to the classical central limit theorem.
Let X1; X2; X3; : : : be i.i.d. real random variables with mean 0 and variance 1, and
let S1; S2; S3; : : : denote the associated partial sums. Let fn denote the density of
Sn=
p
n (if existent), and let ' denote the density of the standard normal distribution.
The classical local limit theorem states that
kfn 'k1 ! 0
if and only if there exists some n0 2 N such that fn0
is bounded.
Since the existence of bounded densities is still a strong condition, it seems natural
to look for similar characterisations under weaker conditions. One such result is the
entropic central limit theorem by Barron (1985), which states that D.fn j '/ ! 0 if
and only if there exists some n0 2 N such that D.fn0
j '/ 1. Here,
D.f j '/ WD
Z
L.f .x/='.x// '.x/dx
denotes the relative entropy of f with respect to ', and L.x/ WD x log x.
A nice interpretation of this result arises from the observation that D.fn j '/ D
H.'/ H.fn/, where H denotes Shannon entropy, and that the standard normal
distribution maximises Shannon entropy among all random variables of mean 0 and
variance 1. Thus, the system converges to the state of maximal entropy. In this
respect, let us also mention the result by Artstein, Ball, Barthe and Naor [5] that
the entropy tends to that of the standard normal distribution monotonously, in line
with the second law of thermodynamics. Moreover, in [6], these authors proved that
the rate of convergence is O.n 1
/ if the random variables Xn satisfy the Poincaré
inequality. However, for more general probability densities, the question of the rate
of convergence remained open.
This question was answered completely in a series of papers by Bobkov,
Chistyakov and Götze [17, 20] from the last few years. Their analysis combined tools
from information theory (such as the entropy convolution inequality) with more clas-
sical results on characteristic functions and their asymptotic approximation. In [17],
the authors derived Edgeworth-type expansions for the entropic central limit theorem.
In particular, if the underlying random variables Xj have finite fourth moments, the
rate of convergence is still O.n 1
/, and hence much better than in the classical cen-
tral limit theorem. This is related to the fact that the term of order O.n 1=2
/ in the
classical asymptotic expansion of the density fn is an odd function, and hence van-
ishes when taking the entropy integral. Moreover, optimal rates for the case where
the random variables Xj have finite fractional moments of order 2 s 4 have
also been obtained [17]. These results rest on technically involved approximations
in the local limit theorem for sums of i.i.d. random variables in the case of fractional
moments. Furthermore, Berry–Esséen bounds in the entropic central limit theorem
have also been established [20].](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-41-320.jpg)
![Convergence and Asymptotic Approximations 23
Similar results were also derived for related notions from information theory, e.g.
Fisher information [18], as well as for other limit laws, e.g. stable distributions [19].
However, in the latter situation some complications arise. For instance, since higher
moments do not exist, asymptotic expansions are not available anymore. Moreover,
the full analogue of the entropic limit theorem can only be obtained for the non-
extremal stable laws. For the extremal stable laws, additional technical conditions are
needed [19]. This might be related to the fact that (non-Gaussian) stable distributions
do not maximise the entropy functional anymore.
Chistyakov and Götze [31] studied related questions for the entropic free central
limit theorem, i.e. they studied the convergence of Voiculescu’s free entropy to its
maximum value (under fixed variance) assumed for the semi-circle distribution. As-
suming a finite moment of order four, they showed that the rate of convergence in
the entropic free CLT is also of the order O.n 1
/. Furthermore, they obtained an ex-
pansion up to an error of order o.n 1
/. These results were based on previous results
about expansions for densities in the free CLT [30].
Besides sums of i.i.d. random variables, maxima of sums of i.i.d. random variables
have also been considered (Bobkov–Chistyakov–Kösters [21]). Here, the limiting
distribution is given by the one-sided normal distribution, which also maximises a
suitably defined entropy functional. Furthermore, one also obtains a characterisation
result here: The maxima of the sums converge to the one-sided normal distribution
in relative entropy if and only if the original random variables Xj have finite relative
entropy with respect to the one-sided normal distribution. The proof is also based on
a combination of the entropy convolution inequality with more classical results for
characteristic functions, including Spitzer’s formula.
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![Convergence and Asymptotic Approximations 27
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![Chapter 2
Kolmogorov operators and SPDEs
M. Röckner
The purpose of this paper is to survey a number of selected results on Kolmogorov
operators and SPDEs. It consists of two parts: One part is related to Kolmogorov
operators and is devoted to the corresponding linear Fokker–Planck–Kolmogorov
equations (see [11]), the other part is about three key results about SPDEs obtained
resp. published during the last funding period, namely an existence and uniqueness
result for L2
-initial data for the stochastic total variation flow, a new approach to
SPDEs and a pathwise uniqueness result of SDEs on Hilbert spaces with a merely
bounded drift part. Both parts are survey type papers coauthored by regular visitors
of the CRC 701. The reader who is interested in a more comprehensive and less se-
lective overview of the results of Project B4 and also Project A9 is referred to the
monographs [1, 5, 11, 19] and the references therein.1
2.1 On recent progress and open problems in the study of linear
Fokker–Planck–Kolmogorov equations
V.I. Bogachev, M. Röckner and S.V. Shaposhnikov
This part reports on some recent progress and remaining challenging open prob-
lems in the study of linear elliptic and parabolic Fokker–Planck–Kolmogorov equa-
tions obtained in Project B4.
2.1.1 Introduction Let us define (linear) Fokker–Planck–Kolmogorov equations in
the elliptic and parabolic cases and formulate several problems related to these equa-
tions. Consider the Kolmogorov operator
Lu.x/ D
d
X
i;j D1
aij
.x/@xi
@xj
u.x/ C
d
X
iD1
bi
.x/@xi
u.x/;
where aij
and bi
are Borel functions on Rd
such that A D .aij
/16i;j 6d is a non-
negative symmetric matrix.
1Projects A9, B4](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-48-320.jpg)
![Kolmogorov operators and SPDEs 31
Let be a locally bounded Borel measure on Rd
. We say that a Borel lo-
cally bounded measure defined by a family of Borel locally bounded measures
.t /0tT satisfies the initial condition
ˇ
ˇ
tD0
D if for every 2 C1
0 .Rd
/
lim
t!0
Z
Rd
.x/ dt D
Z
Rd
.x/ d:
Thus, we study the uniqueness problem for solutions to the Cauchy problem
@t D L
;
ˇ
ˇ
tD0
D ; (2.1.4)
and we are interested in the two classes of solutions: probability and integrable.
A probability solution is a solution defined by a family of probability measures
.t /0tT , i.e., t 0 and t .Rd
/ D 1. The set of all probability solutions D
t .dx/ dt such that jbj 2 L2
.; U Œ0; T / for every ball U Rd
is denoted by P,
where j:j denotes the Euclidean norm on Rd
.
Among many important and interesting problems connected with Fokker–Planck–
Kolmogorov equations and studied in the papers included in the bibliography, we
have chosen for discussion here the following two, related by some similarity in the
nature of assumptions about the equation coefficients: uniqueness of solutions and
bounds on solution densities. Moreover, we confine ourselves to nondegenerate dif-
fusion matrices and, what is especially important, to linear equations. Although we
have also obtained some results on nonlinear equations and degenerate equations,
these directions belong to our future plans.
Before addressing the aforementioned main topics, we briefly recall some basic
facts related to the existence of solution densities and their properties such as local
boundedness and continuity.
It was shown in [10] that, in case of non-negative solutions, the nondegeneracy of
the diffusion matrix at every point is sufficient for the absolute continuity. It is still an
open question whether this is true for signed solutions. For non-negative solutions,
there is a more precise result, which we formulate in the elliptic case: the measure
.detA/1=d
has a density % with respect to Lebesgue measure and % 2 L
d=.d 1/
loc ./.
Therefore, if 1= detA is locally bounded, the measure itself has a density of this
class. This result is sharp in the sense that one cannot omit the factor .det A/1=d
in
front of and also it is impossible to increase the guaranteed order of integrability
d=.d 1/ without additional conditions.
For signed solutions, there is the following result. If the functions aij
belong to
the class VMO (defined below, see also [11]) and on every ball one has 1 Id 6
A.x/ 6 2 Id, where 1; 2 0 are constant, and jbj 2 Lq
loc.Rd
/, where q d, or,
alternatively, jbj 2 L
q
loc.jj/, where q d, then has a density % 2 Lr
loc.Rd
/ for all
numbers r 2 Œ1; C1/. However, these conditions do not ensure local boundedness
of solution densities. Moreover, even the continuity of aij
and bi
along with the
nondegeneracy of A is not enough for that. A sufficient condition on A to ensure the
continuity of densities is stronger: one needs Dini continuity, i.e.,
Z 1
0
!.t/
t
dt 1](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-50-320.jpg)
![32 M. Röckner
for the modulus of continuity ! of A. The result is this. Suppose that, on every ball, A
has a modulus of continuity satisfying the Dini condition. Let det A 0 everywhere
and jbj 2 L
q
loc.Rd
/, where q d. Then, the density % of every solution (possibly,
signed) to the elliptic equation L
D 0 has a continuous version.
If, in addition, the matrix A is locally Hölder continuous of order ı 2 .0; 1/,
then % has a density that is locally Hölder continuous of order ı. Actually, the afore-
mentioned results are proved for equations on domains, but we formulate them for
simplicity in the case of the whole space.
Increasing the regularity of A, one gets more properties for solution densities. If
the matrix elements aij
belong to the local Sobolev class W p;1
loc with p d (which
by the Sobolev embedding theorem yields the existence of a continuous version) and
A is nondegenerate at every point, while the coefficients bi
are either in Lp
loc.Rd
/ or
in Lp
loc.jj/, then the solution density belongs to the same local Sobolev class W p;1
loc .
Therefore, Eq. (2.1.2) can be written in the classical divergence form
@xi
.aij
@xj
%/ C @xj
aij
% bi
% D 0;
(with bi
changed appropriately). Similar results hold in the parabolic case. The
reader is warned that the list of references is far from complete. A vast literature on
the subject is given in the recent monograph [11] presenting results of many authors.
2.1.2 Uniqueness of solutions In the elliptic case, there are examples of infinitely
differentiable b on Rd
with d 1 such that, for A D Id (the unit matrix), there
are several different (hence infinitely many) probability solutions to the equation
L
D 0.
Similarly, in the parabolic case, we have constructed examples of infinitely differ-
entiable b.x/ on Rd
with d 3 such that for A D Id the Cauchy problem with some
initial probability measure has infinitely many probability solutions. Therefore,
certain additional conditions rather than smoothness are needed. We have found con-
ditions of this sort expressed in terms of integrability of the coefficients with respect
to solutions and (an alternative set of assumptions) in terms of so-called Lyapunov
functions. We have also found sufficient conditions for the uniqueness of integrable
solutions. In particular, we have shown that the uniqueness conditions for the class
I of integrable solutions essentially differ from those for the class P. For example,
the following results have been obtained (see [11]).
Theorem 2.1.1. The elliptic equation L
D 0 has at most one probability solution
in either of the following cases, assuming in these cases (for simplicity) that A is
locally Lipschitz and b is locally bounded.
(i) There is a non-negative function V 2 C2
.Rd
/ such that lim
jxj!1
V.x/ D C1
and
LV.x/ 6 CV.x/ for some C 0.
(ii)
aij
1 C jxj2
;
bi
1 C jxj
2 L1
./ for some probability solution .](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-51-320.jpg)
![Kolmogorov operators and SPDEs 35
with the operator
L D
1
X
i;j D1
aij
@xi
@xj
C
1
X
iD1
bi
@xi
if the functions aij
, bi
are -integrable and for every function ' in finitely many
variables belonging to the corresponding class C1
b
one has
Z
R1
L' d D 0:
Obviously, the function L' is -integrable since the series becomes a finite sum of
-integrable functions. Set b WD .bi
/1
iD1.
Similarly, one defines a parabolic equation and the corresponding Cauchy prob-
lem.
As in the finite-dimensional case, interesting problems arise already for aij
D ıij
(which in Rd
would mean the unit diffusion matrix). For example, if b.x/ D x, the
standard Gaussian measure on R1
(the countable power of the standard Gaussian
measure on the real line) is a solution to the corresponding equation. This measure is
the only probability solution. Indeed, for a general b, the projection of any solution
to Rn
, denoted by n, satisfies the equation on Rn
whose diffusion coefficients
aij
n and drift coefficients bi
n are the conditional expectations of the functions aij
and
bi
with i; j 6 n with respect to the measure and the -algebra Bn generated
by the first n coordinates. This is obvious from the definitions. Therefore, in case
of constant aij
, we have aij
n D aij
and, in case of bi
with i 6 n depending on
x1; : : : ; xn, we have bi
n D bi
. In particular, for bi
.x/ D xi , we have bi
n.x/ D xi
whenever i 6 n. Therefore, in the situation under consideration, the projection of
any probability solution to Rn
satisfies the same equation as the standard Gaussian
measure on Rn
, but this equation admits only one probability solution.
However, the situation may change for other b. Actually, examples of linear b
are known such that the corresponding equation (with aij
D ıij
) has several different
probability solutions that are Gaussian measures (hence there exist also non-Gaussian
solutions, their convex combinations).
A relatively simple case arises if we take
b.x/ D x C v.x/;
where the perturbation v takes values in the Hilbert space H D l2
and is bounded in
the usual l2
-norm. In this case, it is known that every probability solution to the
elliptic equation L
D 0 is absolutely continuous with respect to and is unique
(it is also known that a solution exists under the stated assumptions). Certainly, the
assumption that v is a bounded l2
-valued perturbation is very restrictive.
As a typical result on uniqueness in infinite dimensions, we mention the following
theorem from [9]. Let us consider the following Cauchy problem.
Let B D .Bi
/ be a sequence of Borel functions on R1
.0; T0/, where T0 0
is fixed, and let aij
be Borel functions on R1
.0; T0/. Let us consider the Cauchy](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-54-320.jpg)
![[41] English As We Speak It In Ireland, p. 77.
[42] The Science of Language, vol. i, p. 166.
[43] The last stand of the distinct -ed was made in Addison's day. He was in
favor of retaining it, and in the Spectator for Aug. 4, 1711, he protested
against obliterating the syllable in the termination of our praeter perfect
tense, as in these words, drown'd, walk'd, arriv'd, for drowned, walked,
arrived, which has very much disfigured the tongue, and turned a tenth
part of our smoothest words into so many clusters of consonants.
[44] A New English Grammar, pt. i, p. 380.
[45] History of the English Language, p. 398.
[46] And still more often as an adjective, as in it was a boughten dress.
[47] You Know Me Al, p. 180; see also p. 122.
[48] Cf. Lounsbury: History of the English Language, pp. 393 et seq.
[49] Remark of a policeman talking to another. What he actually said was
before the Elks was c'm 'ere. Come and here were one word,
approximately cmear. The context showed that he meant to use the past
perfect tense.
[50] These examples are from Lardner's story, A New Busher Breaks In, in
You Know Me Al, pp. 122 et seq.
[51] You Know Me Al, op. cit., p. 124.
[52] The Making of English, p. 53.
[53] Cf. Dialect Notes, vol. iii, pt. i, p. 59; ibid., vol. III, pt. iv, p. 283.
[54] Henry Bradley, in The Making of English, pp. 54-5: In the parts of
England which were largely inhabited by Danes the native pronouns
(i. e., heo, his, heom and heora) were supplanted by the Scandinavian
pronouns which are represented by the modern she, they, them and
their. This substitution, at first dialectical, gradually spread to the whole
language.
[55] Cf. Sweet: A New English Grammar, pt. i, p. 344, par. 1096.
[56] Before a noun beginning with a vowel thine and mine are commonly
substituted for thy and my, as in thine eyes and mine infirmity. But
this is solely for the sake of euphony. There is no compensatory use of
my and thy in the absolute.
[57] The Making of English, p. 58.
[58] Cf. The Dialect of Southeastern Missouri, by D. S. Crumb, Dialect Notes,
vol. ii, pt. iv, 1903, p. 337.
[59] It occurs, too, of course, in other dialects of English, though by no
means in all. The Irish influence probably had something to do with its](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-56-320.jpg)
![prosperity in vulgar American. At all events, the Irish use it in the
American manner. Joyce, in English As We Speak It in Ireland, pp. 34-5,
argues that this usage was suggested by Gaelic. In Gaelic the accusative
pronouns, e, i and iad (=him, her and them) are often used in place of
the nominatives, sé, si and siad (=he, she and they), as in is iad sin na
buachaillidhe (=them are the boys). This is good grammar in Gaelic,
and the Irish, when they began to learn English, translated the locution
literally. The familiar Irish John is dead and him always so hearty
shows the same influence.
[60] Pp. 144-50.
[61] Modern English, p. 300.
[62] A New English Grammar, pt. i, p. 339.
[63] History of the English Language, pp. 274-5.
[64] Modern English, p. 288-9.
[65] Cf. p. 145n.
[66] A New English Grammar, pt. i, p. 341.
[67] It may be worth noting here that the misuse of me for my, as in I lit
me pipe is quite unknown in American, either standard or vulgar. Even
me own is seldom heard. This boggling of the cases is very common in
spoken English.
[68] A New English Grammar, pt. i, p. 341.
[69] The King's English, p. 63.
[70] Hon. Edward E. Browne, of Wisconsin, in the House of
Representatives, July 18, 1918, p. 9965.
[71] Cf. Vogue Affixes in Present-Day Word-Coinage, by Louise Pound, Dialect
Notes, vol. v, pt. i, 1918.
[72] The Speech of a Child Two Years of Age, Dialect Notes, vol. iv, pt. ii,
1914.
[73] A New English Grammar, pt. i, pp. 437-8.
[74] The King's English, p. 322. See especially the quotation from Frederick
Greenwood, the distinguished English journalist.
[75] Report of Edward J. Brundage, attorney-general of Illinois, on the East
St. Louis massacre, Congressional Record, Jan. 7, 1918, p. 661.
[76] The King's English, op. cit.
[77] Oct. 1, 1864.
[78] At all, by the way, is often displaced by any or none, as in he don't
lover her any and it didn't hurt me none.
[79] See the bibliography for the publication of Drs. Read and Pound.](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-57-320.jpg)
![[80] The only book that I can find definitely devoted to American sounds is A
Handbook of American Speech, by Calvin L. Lewis; Chicago, 1916. It has
many demerits. For example, the author gives a z-sound to the s in
venison (p. 52). This is surely not American.
[81] Maryland edition, July 18, 1914, p. 1.
[82] Cf. Lounsbury: The Standard of Pronunciation in English, p. 172 et seq.
[83] Stomp is used only in the sense of to stamp with the foot. One always
stamps a letter. An analogue of stomp, accepted in correct English, is
strop (e. g., razor-strop), from strap.
[84] Our Own, Our Native Speech, McClure's Magazine, Oct., 1916.
[Pg242] toc](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-58-320.jpg)
![caliber calibre
candor candour
center centre
check (bank) cheque
checkered chequered
cider cyder
clamor clamour
clangor clangour
cloture closure[1]
color colour
connection connexion
councilor councillor
counselor counsellor
cozy cosy
curb kerb
cyclopedia cyclopaedia
defense defence
demeanor demeanour
diarrhea diarrhoea
draft (ship's) draught
dreadnaught dreadnought
dryly drily
ecology oecology
ecumenical oecumenical
edema oedema
encyclopedia encyclopaedia
endeavor endeavour
eon aeon
epaulet epaulette
esophagus oesophagus](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-60-320.jpg)
![skeptic sceptic
slug (verb) slog
slush slosh
splendor splendour
stanch staunch
story (of a house) storey
succor succour
taffy toffy
tire (noun) tyre
toilet toilette
traveler traveller
tumor tumour
valor valour
vapor vapour
veranda verandah
vial phial
vigor vigour
vise (a tool) vice
wagon waggon
woolen woollen
§ 2
General Tendencies
—This list is by no means exhaustive. According to a recent writer
upon the subject, there are 812 words in which the prevailing
American spelling differs from the English.[2] But enough examples
are given to reveal a number of definite tendencies. American, in
general, moves toward simplified forms of spelling more rapidly than
English, and has got much further along the road. Redundant and](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-63-320.jpg)
![unnecessary letters have been dropped from whole groups of words
—the u from the group of nouns in -our, with the sole exception of
Saviour, and from such words as mould and baulk; the e from
annexe, asphalte, axe, forme, pease, storey, etc.; the duplicate
consonant from waggon, nett, faggot, woollen, jeweller, councillor,
etc., and the silent foreign suffixes from toilette, epaulette,
programme, verandah, etc. In addition, simple vowels have been
substituted for degenerated diphthongs in such words as anaemia,
[Pg246] oesophagus, diarrhoea and mediaeval, most of them from the
Greek.
Further attempts in the same direction are to be seen in the
substitution of simple consonants for compound consonants, as in
plow, bark, check, vial and draft; in the substitution of i for y to
bring words into harmony with analogues, as in tire, cider and
baritone (cf. wire, rider, merriment), and in the general tendency to
get rid of the somewhat uneuphonious y, as in ataxia and pajamas.
Clarity and simplicity are also served by substituting ct for x in such
words as connection and inflection, and s for c in words of the
defense group. The superiority of jail to gaol is made manifest by
the common mispronunciation of the latter, making it rhyme with
coal. The substitution of i for e in such words as indorse, inclose and
jimmy is of less patent utility, but even here there is probably a
slight gain in euphony. Of more obscure origin is what seems to be a
tendency to avoid the o-sound, so that the English slog becomes
slug, podgy becomes pudgy, nought becomes naught, slosh
becomes slush, toffy becomes taffy, and so on. Other changes carry
their own justification. Hostler is obviously better American than
ostler, though it may be worse English. Show is more logical than
shew.[3] Cozy is more nearly phonetic than cosy. Curb has analogues
in curtain, curdle, curfew, curl, currant, curry, curve, curtsey, curse,
currency, cursory, curtail, cur, curt and many other common words:
kerb has very few, and of them only kerchief and kernel are in
general use. Moreover, the English themselves use curb as a verb
and in all noun senses save that shown in kerbstone.](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-64-320.jpg)
![But a number of anomalies remain. The American substitution of a
for e in gray is not easily explained, nor is the substitution of k for c
in skeptic and mollusk, nor the retention of e in forego, nor the
unphonetic substitution of s for z in fuse, [Pg247] nor the persistence
of the first y in pygmy. Here we have plain vagaries, surviving in
spite of attack by orthographers. Webster, in one of his earlier
books, denounced the k in skeptic as a mere pedantry, but later on
he adopted it. In the same way pygmy, gray and mollusk have been
attacked, but they still remain sound American. The English
themselves have many more such illogical forms to account for. In
the midst of the our-words they cling to a small number in or,
among them, stupor. Moreover, they drop the u in many derivatives,
for example, in arboreal, armory, clamorously, clangorous,
odoriferous, humorist, laborious and rigorism. If it were dropped in
all derivatives the rule would be easy to remember, but it is retained
in some of them, for example, colourable, favourite, misdemeanour,
coloured and labourer. The derivatives of honour exhibit the
confusion clearly. Honorary, honorarium and honorific drop the u,
but honourable retains it. Furthermore, the English make a
distinction between two senses of rigor. When used in its
pathological sense (not only in the Latin form of rigor mortis, but as
an English word) it drops the u; in all other senses it retains the u.
The one American anomaly in this field is Saviour. In its theological
sense it retains the u; but in that sense only. A sailor who saves his
ship is its savior, not its saviour.
§ 3
The Influence of Webster
—At the time of the first settlement of America the rules of English
orthography were beautifully vague, and so we find the early
documents full of spellings that would give an English lexicographer
much pain today. Now and then a curious foreshadowing of later
American usage is encountered. On July 4, 1631, for example, John](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-65-320.jpg)
![Winthrop wrote in his journal that the governour built a bark at
Mistick, which was launched this day. But during the eighteenth
century, and especially after the publication of Johnson's dictionary,
there was a general movement in England toward a more inflexible
orthography, and many hard and fast rules, still surviving, were then
laid down. It was Johnson himself who [Pg248] established the position
of the u in the our words. Bailey, Dyche and the other lexicographers
before him were divided and uncertain; Johnson declared for the u,
and though his reasons were very shaky[4] and he often neglected
his own precept, his authority was sufficient to set up a usage which
still defies attack in England. Even in America this usage was not
often brought into question until the last quarter of the eighteenth
century. True enough, honor appears in the Declaration of
Independence, but it seems to have got there rather by accident
than by design. In Jefferson's original draft it is spelled honour. So
early as 1768 Benjamin Franklin had published his Scheme for a
New Alphabet and a Reformed Mode of Spelling, with Remarks and
Examples Concerning the Same, and an Enquiry Into its Uses and
induced a Philadelphia typefounder to cut type for it, but this
scheme was too extravagant to be adopted anywhere, or to have
any appreciable influence upon spelling.[5]
It was Noah Webster who finally achieved the divorce between
English example and American practise. He struck the first blow in
his Grammatical Institute of the English Language, published at
Hartford in 1783. Attached to this work was an appendix bearing the
formidable title of An Essay on the Necessity, Advantages and
Practicability of Reforming the Mode of Spelling, and of Rendering
the Orthography of Words Correspondent to the Pronunciation, and
during the same year, at Boston, he set forth his ideas a second time
in the first edition of his American Spelling Book. The influence of
this spelling book was immediate and profound. It took the place in
the schools of Dilworth's Aby-sel-pha, the favorite of the
generation preceding, and maintained its authority for fully a
century. Until Lyman Cobb entered the lists with his New Spelling
Book, in 1842, its innumerable editions scarcely had [Pg249] any](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-66-320.jpg)
![rivalry, and even then it held its own. I have a New York edition,
dated 1848, which contains an advertisement stating that the annual
sale at that time was more than a million copies, and that more than
30,000,000 copies had been sold since 1783. In the late 40's the
publishers, George F. Cooledge Bro., devoted the whole capacity of
the fastest steam press in the United States to the printing of it. This
press turned out 525 copies an hour, or 5,250 a day. It was
constructed expressly for printing Webster's Elementary Spelling
Book [the name had been changed in 1829] at an expense of
$5,000. Down to 1889, 62,000,000 copies of the book had been
sold.
The appearance of Webster's first dictionary, in 1806, greatly
strengthened his influence. The best dictionary available to
Americans before this was Johnson's in its various incarnations, but
against Johnson's stood a good deal of animosity to its compiler,
whose implacable hatred of all things American was well known to
the citizens of the new republic. John Walker's dictionary, issued in
London in 1791, was also in use, but not extensively. A home-made
school dictionary, issued at New Haven in 1798 or 1799 by one
Samuel Johnson, Jr.—apparently no relative of the great Sam—and a
larger work published a year later by Johnson and the Rev. John
Elliott, pastor in East Guilford, Conn., seem to have made no
impression, despite the fact that the latter was commended by
Simeon Baldwin, Chauncey Goodrich and other magnificoes of the
time and place, and even by Webster himself. The field was thus
open to the laborious and truculent Noah. He was already the
acknowledged magister of lexicography in America, and there was
an active public demand for a dictionary that should be wholly
American. The appearance of his first duodecimo, according to
Williams,[6] thereby took on something of the character of a national
event. It was received, not critically, but patriotically, and its
imperfections were swallowed as eagerly as its merits. Later on
Webster had to meet formidable critics, at home as well as abroad,
but for nearly a quarter of a century he reigned almost
unchallenged. Edition after edition of his dictionary was published,](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-67-320.jpg)
![[Pg250] each new one showing additions and improvements. Finally, in
1828, he printed his great American Dictionary of the English
Language, in two large octavo volumes. It held the field for half a
century, not only against Worcester and the other American
lexicographers who followed him, but also against the best
dictionaries produced in England. Until very lately, indeed, America
remained ahead of England in practical dictionary making.
Webster had declared boldly for simpler spellings in his early
spelling books; in his dictionary of 1806 he made an assault at all
arms upon some of the dearest prejudices of English lexicographers.
Grounding his wholesale reforms upon a saying by Franklin, that
those people spell best who do not know how to spell—i. e., who
spell phonetically and logically—he made an almost complete sweep
of whole classes of silent letters—the u in the -our words, the final e
in determine and requisite, the silent a in thread, feather and
steady, the silent b in thumb, the s in island, the o in leopard, and
the redundant consonants in traveler, wagon, jeweler, etc. (English:
traveller, waggon, jeweller). More, he lopped the final k from frolick,
physick and their analogues. Yet more, he transposed the e and the
r in all words ending in re, such as theatre, lustre, centre and
calibre. Yet more, he changed the c in all words of the defence class
to s. Yet more, he changed ph to f in words of the phantom class, ou
to oo in words of the group class, ow to ou in crowd, porpoise to
porpess, acre to aker, sew to soe, woe to wo, soot to sut, gaol to
jail, and plough to plow. Finally, he antedated the simplified spellers
by inventing a long list of boldly phonetic spellings, ranging from
tung for tongue to wimmen for women, and from hainous for
heinous to cag for keg.
A good many of these new spellings, of course, were not actually
Webster's inventions. For example, the change from -our to -or in
words of the honor class was a mere echo of an earlier English
usage, or, more accurately, of an earlier English uncertainty. In the
first three folios of Shakespeare, 1623, 1632 and 1663-6, honor and
honour were used indiscriminately and in almost equal proportions;
English spelling was still fluid, and [Pg251] the -our-form was not](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-68-320.jpg)
![consistently adopted until the fourth folio of 1685. Moreover, John
Wesley, the founder of Methodism, is authority for the statement
that the -or-form was a fashionable impropriety in England in
1791. But the great authority of Johnson stood against it, and
Webster was surely not one to imitate fashionable improprieties. He
deleted the u for purely etymological reasons, going back to the
Latin honor, favor and odor without taking account of the
intermediate French honneur, faveur and odeur. And where no
etymological reasons presented themselves, he made his changes by
analogy and for the sake of uniformity, or for euphony or simplicity,
or because it pleased him, one guesses, to stir up the academic
animals. Webster, in fact, delighted in controversy, and was anything
but free from the national yearning to make a sensation.
A great many of his innovations, of course, failed to take root, and
in the course of time he abandoned some of them himself. In his
early Essay on the Necessity, Advantage and Practicability of
Reforming the Mode of Spelling he advocated reforms which were
already discarded by the time he published the first edition of his
dictionary. Among them were the dropping of the silent letter in such
words as head, give, built and realm, making them hed, giv, bilt and
relm; the substitution of doubled vowels for decayed diphthongs in
such words as mean, zeal and near, making them meen, zeel and
neer; and the substitution of sh for ch in such French loan-words as
machine and chevalier, making them masheen and shevaleer. He
also declared for stile in place of style, and for many other such
changes, and then quietly abandoned them. The successive editions
of his dictionary show still further concessions. Croud, fether, groop,
gillotin, iland, insted, leperd, soe, sut, steddy, thret, thred, thum and
wimmen appear only in the 1806 edition. In 1828 he went back to
crowd, feather, group, island, instead, leopard, sew, soot, steady,
thread, threat, thumb and women, and changed gillotin to guillotin.
In addition, he restored the final e in determine, discipline, requisite,
imagine, etc. In 1838, revising his dictionary, he abandoned a good
many spellings that had appeared in either the 1806 or the 1828
edition, notably maiz for maize, [Pg252] suveran for sovereign and](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-69-320.jpg)
![guillotin for guillotine. But he stuck manfully to a number that were
quite as revolutionary—for example, aker for acre, cag for keg,
grotesk for grotesque, hainous for heinous, porpess for porpoise and
tung for tongue—and they did not begin to disappear until the
edition of 1854, issued by other hands and eleven years after his
death. Three of his favorites, chimist for chemist, neger for negro
and zeber for zebra, are incidentally interesting as showing changes
in American pronunciation. He abandoned zeber in 1828, but
remained faithful to chimist and neger to the last.
But though he was thus forced to give occasional ground, and in
more than one case held out in vain, Webster lived to see the
majority of his reforms adopted by his countrymen. He left the
ending in -or triumphant over the ending in -our, he shook the
security of the ending in -re, he rid American spelling of a great
many doubled consonants, he established the s in words of the
defense group, and he gave currency to many characteristic
American spellings, notably jail, wagon, plow, mold and ax. These
spellings still survive, and are practically universal in the United
States today; their use constitutes one of the most obvious
differences between written English and written American. Moreover,
they have founded a general tendency, the effects of which reach far
beyond the field actually traversed by Webster himself. New words,
and particularly loan-words, are simplified, and hence naturalized in
American much more quickly than in English. Employé has long since
become employee in our newspapers, and asphalte has lost its final
e, and manoeuvre has become maneuver, and pyjamas has become
pajamas. Even the terminology of science is simplified and
Americanized. In medicine, for example, the highest American usage
countenances many forms which would seem barbarisms to an
English medical man if he encountered them in the Lancet. In
derivatives of the Greek haima it is the almost invariable American
custom to spell the root syllable hem, but the more conservative
English make it haem—e. g., in haemorrhage and haemiplegia. In an
exhaustive list of diseases issued by the United States Public Health
[Pg253] Service[7] the haem-form does not appear once. In the same](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-70-320.jpg)
![way American usage prefers esophagus, diarrhea and gonorrhea to
the English oesophagus, diarrhoea and gonorrhoea. In the style-
book of the Journal of the American Medical Association[8] I find
many other spellings that would shock an English medical author,
among them curet for curette, cocain for cocaine, gage for gauge,
intern for interne, lacrimal for lachrymal, and a whole group of
words ending in -er instead of in -re.
Webster's reforms, it goes without saying, have not passed
unchallenged by the guardians of tradition. A glance at the literature
of the first years of the nineteenth century shows that most of the
serious authors of the time ignored his new spellings, though they
were quickly adopted by the newspapers. Bancroft's Life of
Washington contains -our endings in all such words as honor, ardor
and favor. Washington Irving also threw his influence against the -or
ending, and so did Bryant and most of the other literary big-wigs of
that day. After the appearance of An American Dictionary of the
English Language, in 1828, a formal battle was joined, with Lyman
Cobb and Joseph E. Worcester as the chief opponents of the
reformer. Cobb and Worcester, in the end, accepted the -or ending
and so surrendered on the main issue, but various other champions
arose to carry on the war. Edward S. Gould, in a once famous essay,
[9] denounced the whole Websterian orthography with the utmost
fury, and Bryant, reprinting this philippic in the Evening Post, said
that on account of Webster the English language has been
undergoing a process of corruption for the last quarter of a century,
and offered to contribute to a fund to have Gould's denunciation
read twice a year in every school-house in the United States, until
every trace of Websterian spelling disappears from the land. But
Bryant was forced to admit that, even in 1856, the chief novelties of
the Connecticut school-master who taught millions to read but not
one to sin were [Pg254] adopted and propagated by the largest
publishing house, through the columns of the most widely circulated
monthly magazine, and through one of the ablest and most widely
circulated newspapers in the United States—which is to say, the
Tribune under Greeley. The last academic attack was delivered by](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-71-320.jpg)
![Bishop Coxe in 1886, and he contented himself with the resigned
statement that Webster has corrupted our spelling sadly.
Lounsbury, with his active interest in spelling reform, ranged himself
on the side of Webster, and effectively disposed of the controversy
by showing that the great majority of his spellings were supported
by precedents quite as respectable as those behind the fashionable
English spellings. In Lounsbury's opinion, a good deal of the
opposition to them was no more than a symptom of antipathy to all
things American among certain Englishmen and of subservience to
all things English among certain Americans.[10]
Webster's inconsistency gave his opponents a formidable weapon
for use against him—until it began to be noticed that the orthodox
English spelling was quite as inconsistent. He sought to change acre
to aker, but left lucre unchanged. He removed the final f from bailiff,
mastiff, plaintiff and pontiff, but left it in distaff. He changed c to s in
words of the offense class, but left the c in fence. He changed the ck
in frolick, physick, etc., into a simple c, but restored it in such
derivatives as frolicksome. He deleted the silent u in mould, but left
it in court. These slips were made the most of by Cobb in a
pamphlet printed in 1831.[11] He also detected Webster in the
frequent faux pas of using spellings in his definitions and
explanations that conflicted with the spellings he advocated. Various
other purists joined in the attack, and it was renewed with great fury
after the appearance of Worcester's dictionary, in 1846. Worcester,
who had begun his lexicographical labors by editing Johnson's
dictionary, was a good deal more conservative than Webster, and so
the partisans of conformity rallied around him, and for [Pg255] a while
the controversy took on all the rancor of a personal quarrel. Even
the editions of Webster printed after his death, though they gave
way on many points, were violently arraigned. Gould, in 1867,
belabored the editions of 1854 and 1866,[12] and complained that
for the past twenty-five years the Websterian replies have uniformly
been bitter in tone, and very free in the imputation of personal
motives, or interested or improper motives, on the part of opposing
critics. At this time Webster himself had been dead for twenty-two](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-72-320.jpg)
![years. Schele de Vere, during the same year, denounced the
publishers of the Webster dictionaries for applying immense capital
and a large stock of energy and perseverance to the propagation of
his new and arbitrarily imposed orthography.[13]
§ 4
Exchanges
—As in vocabulary and in idiom, there are constant exchanges
between English and American in the department of orthography.
Here the influence of English usage is almost uniformly toward
conservatism, and that of American usage is as steadily in the other
direction. The logical superiority of American spelling is well
exhibited by its persistent advance in the face of the utmost hostility.
The English objection to our simplifications, as Brander Matthews
points out, is not wholly or even chiefly etymological; its roots lie, to
borrow James Russell Lowell's phrase, in an esthetic hatred burning
with as fierce a flame as ever did theological hatred. There is
something inordinately offensive to English purists in the very
thought of taking lessons from this side of the water, particularly in
the mother tongue. The opposition, transcending the academic,
takes on the character of the patriotic. Any American, continues
Matthews, who chances to note the force and the fervor and the
frequency of the objurgations against American spelling in the
columns of the Saturday Review, for example, and of the
Athenaeum, may find himself wondering as to the date of the [Pg256]
papal bull which declared the infallibility of contemporary British
orthography, and as to the place where the council of the Church
was held at which it was made an article of faith.[14] This was
written more than a quarter of a century ago. Since then there has
been a lessening of violence, but the opposition still continues. No
self-respecting English author would yield up the -our ending for an
instant, or write check for cheque, or transpose the last letters in the
-re words.](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-73-320.jpg)
![Nevertheless, American spelling makes constant gains across the
water, and they more than offset the occasional fashions for English
spellings on this side. Schele de Vere, in 1867, consoled himself for
Webster's arbitrarily imposed orthography by predicting that it
could be only temporary—that, in the long run, North America
depends exclusively on the mother-country for its models of
literature. But the event has blasted this prophecy and confidence,
for the English, despite their furious reluctance, have succumbed to
Webster more than once. The New English Dictionary, a monumental
work, shows many silent concessions, and quite as many open
yieldings—for example, in the case of ax, which is admitted to be
better than axe on every ground. Moreover, English usage tends to
march ahead of it, outstripping the liberalism of its editor, Sir James
A. H. Murray. In 1914, for example, Sir James was still protesting
against dropping the first e from judgement, a characteristic
Americanism, but during the same year the Fowlers, in their Concise
Oxford Dictionary, put judgment ahead of judgement; and two years
earlier the Authors' and Printers' Dictionary, edited by Horace Hart,
[15] had dropped judgement altogether. Hart is Controller of the
Oxford University Press, and the Authors' and Printers' Dictionary is
an authority accepted by nearly all of the great English book
publishers and newspapers. Its last edition shows a great many
American spellings. For example, it recommends the use of jail and
jailer in place [Pg257] of the English gaol and gaoler, says that ax is
better than axe, drops the final e from asphalte and forme, changes
the y to i in cyder, cypher and syren and advocates the same change
in tyre, drops the redundant t from nett, changes burthen to burden,
spells wagon with one g, prefers fuse to fuze, and takes the e out of
storey. Rules for Compositors and Readers at the University Press,
Oxford, also edited by Hart (with the advice of Sir James Murray
and Dr. Henry Bradley), is another very influential English authority.
[16] It gives its imprimatur to bark (a ship), cipher, siren, jail, story,
tire and wagon, and even advocates kilogram and omelet. Finally,
there is Cassell's English Dictionary.[17] It clings to the -our and -re
endings and to annexe, waggon and cheque, but it prefers jail to](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-74-320.jpg)
![gaol, net to nett, asphalt to asphalte and story to storey, and comes
out flatly for judgment, fuse and siren.
Current English spelling, like our own, shows a number of
uncertainties and inconsistencies, and some of them are
undoubtedly the result of American influences that have not yet
become fully effective. The lack of harmony in the -our words,
leading to such discrepancies as honorary and honourable, I have
already mentioned. The British Board of Trade, in attempting to fix
the spelling of various scientific terms, has often come to grief. Thus
it detaches the final -me from gramme in such compounds as
kilogram and milligram, but insists upon gramme when the word
stands alone. In American usage gram is now common, and scarcely
challenged. All the English authorities that I have consulted prefer
metre and calibre to the American meter and caliber.[18] They also
support the ae in such words as aetiology, aesthetics, mediaeval and
anaemia, and the oe in oesophagus, [Pg258] manoeuvre and
diarrhoea. They also cling to such forms as mollusc, kerb, pyjamas
and ostler, and to the use of x instead of ct in connexion and
inflexion. The Authors' and Printers' Dictionary admits the American
curb, but says that the English kerb is more common. It gives
barque, plough and fount, but grants that bark, plow and font are
good in America. As between inquiry and enquiry, it prefers the
American inquiry to the English enquiry, but it rejects the American
inclose and indorse in favor of the English enclose and endorse.[19]
Here American spelling has driven in a salient, but has yet to take
the whole position. A number of spellings, nearly all American, are
trembling on the brink of acceptance in both countries. Among them
is rime (for rhyme). This spelling was correct in England until about
1530, but its recent revival was of American origin. It is accepted by
the Oxford Dictionary and by the editors of the Cambridge History of
English Literature, but it seldom appears in an English journal. The
same may be said of grewsome. It has got a footing in both
countries, but the weight of English opinion is still against it. Develop
(instead of develope) has gone further in both countries. So has
engulf, for engulph. So has gipsy for gypsy.](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-75-320.jpg)
![American imitation of English orthography has two impulses
behind it. First, there is the colonial spirit, the desire to pass as
English—in brief, mere affectation. Secondly, there is the wish
among printers, chiefly of books and periodicals, to reach a
compromise spelling acceptable in both countries, thus avoiding
expensive revisions in case of republication in England.[20] [Pg259] The
first influence need not detain us. It is chiefly visible among folk of
fashionable pretensions, and is not widespread. At Bar Harbor, in
Maine, some of the summer residents are at great pains to put
harbour instead of harbor on their stationery, but the local
postmaster still continues to stamp all mail Bar Harbor, the legal
name of the place. In the same way American haberdashers
sometimes advertise pyjamas instead of pajamas, just as they
advertise braces instead of suspenders and vests instead of
undershirts. But this benign folly does not go very far. Beyond
occasionally clinging to the -re ending in words of the theatre group,
all American newspapers and magazines employ the native
orthography, and it would be quite as startling to encounter honour
or jewellery in one of them as it would be to encounter gaol or
waggon. Even the most fashionable jewelers in Fifth avenue still deal
in jewelry, not in jewellery.
The second influence is of more effect and importance. In the
days before the copyright treaty between England and the United
States, one of the standing arguments against it among the English
was based upon the fear that it would flood England with books set
up in America, and so work a corruption of English spelling.[21] This
fear, as we have seen, had a certain plausibility; there is not the
slightest doubt that American books and American magazines have
done valiant missionary service for American orthography. But
English conservatism still holds out stoutly enough to force American
printers to certain compromises. When a book is designed for
circulation in both countries it is common for the publisher to instruct
the printer to employ English spelling. This English spelling, at the
Riverside Press,[22] embraces all the -our endings and the following
further forms:](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-76-320.jpg)
![cheque
chequered
connexion
dreamt
faggot
forgather
forgo
grey
inflexion
jewellery
leapt
premises (in logic)
waggon
It will be noted that gaol, tyre, storey, kerb, asphalte, annexe,
ostler, mollusc and pyjamas are not listed, nor are the words ending
in -re. These and their like constitute the English contribution to the
compromise. Two other great American book presses, that of the
Macmillan Company[23] and that of the J. S. Cushing Company,[24]
add gaol and storey to the list, and also behove, briar, drily, enquire,
gaiety, gipsy, instal, judgement, lacquey, moustache, nought, pigmy,
postillion, reflexion, shily, slily, staunch and verandah. Here they go
too far, for, as we have seen, the English themselves have begun to
abandon briar, enquire and judgement. Moreover, lacquey is going
out over there, and gipsy is not English, but American. The Riverside
Press, even in books intended only for America, prefers certain
English forms, among them, anaemia, axe, mediaeval, mould,
plough, programme and quartette, but in compensation it stands by
such typical Americanisms as caliber, calk, center, cozy, defense,
foregather, gray, hemorrhage, luster, maneuver, mustache, theater](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-77-320.jpg)
![and woolen. The Government Printing Office at Washington follows
Webster's New International Dictionary,[25] which supports most of
the innovations of Webster himself. This dictionary is the authority in
perhaps a majority of American printing offices, with the Standard
and the Century supporting it. The latter two also follow Webster,
notably in his -er [Pg261] endings and in his substitution of s for c in
words of the defense class. The Worcester Dictionary is the sole
exponent of English spelling in general circulation in the United
States. It remains faithful to most of the -re endings, and to
manoeuvre, gramme, plough, sceptic, woollen, axe and many other
English forms. But even Worcester favors such characteristic
American spellings as behoove, brier, caliber, checkered, dryly, jail
and wagon.
§ 5
Simplified Spelling
—The current movement toward a general reform of English-
American spelling is of American origin, and its chief supporters are
Americans today. Its actual father was Webster, for it was the long
controversy over his simplified spellings that brought the dons of the
American Philological Association to a serious investigation of the
subject. In 1875 they appointed a committee to inquire into the
possibility of reform, and in 1876 this committee reported favorably.
During the same year there was an International Convention for the
Amendment of English Orthography at Philadelphia, with several
delegates from England present, and out of it grew the Spelling
Reform Association.[26] In 1878 a committee of American philologists
began preparing a list of proposed new spellings, and two years
later the Philological Society of England joined in the work. In 1883 a
joint manifesto was issued, recommending various general
simplifications. In 1886 the American Philological Association issued
independently a list of recommendations affecting about 3,500
words, and falling under ten headings. Practically all of the changes](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-78-320.jpg)
![proposed had been put forward 80 years before by Webster, and
some of them had entered into unquestioned American usage in the
meantime, e. g., the deletion of the u from the -our words, the
substitution of [Pg262] er for re at the end of words, the reduction of
traveller to traveler, and the substitution of z for s wherever
phonetically demanded, as in advertize and cozy.
The trouble with the others was that they were either too uncouth
to be adopted without a struggle or likely to cause errors in
pronunciation. To the first class belonged tung for tongue, ruf for
rough, batl for battle and abuv for above, and to the second such
forms as cach for catch and troble for trouble. The result was that
the whole reform received a set-back: the public dismissed the
industrious professors as a pack of dreamers. Twelve years later the
National Education Association revived the movement with a
proposal that a beginning be made with a very short list of reformed
spellings, and nominated the following by way of experiment: tho,
altho, thru, thruout, thoro, thoroly, thorofare, program, prolog,
catalog, pedagog and decalog. This scheme of gradual changes was
sound in principle, and in a short time at least two of the
recommended spellings, program and catalog, were in general use.
Then, in 1906, came the organization of the Simplified Spelling
Board, with an endowment of $15,000 a year from Andrew
Carnegie, and a formidable membership of pundits. The board at
once issued a list of 300 revised spellings, new and old, and in
August, 1906, President Roosevelt ordered their adoption by the
Government Printing Office. But this unwise effort to hasten matters,
combined with the buffoonery characteristically thrown about the
matter by Roosevelt, served only to raise up enemies, and since
then, though it has prudently gone back to more discreet endeavors
and now lays main stress upon the original 12 words of the National
Education Association, the Board has not made a great deal of
progress.[27] From time to time it issues impressive lists of
newspapers and periodicals that are using some, at least, of its
revised spellings and of colleges that have made them optional, but
an inspection of these lists shows that very few [Pg263] publications of](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-79-320.jpg)
![any importance have been converted[28] and that most of the great
universities still hesitate. It has, however, greatly reinforced the
authority behind many of Webster's spellings, and it has done much
to reform scientific orthography. Such forms as gram, cocain, chlorid,
anemia and anilin are the products of its influence.
Despite the large admixture of failure in this success there is good
reason to believe that at least two of the spellings on the National
Education Association list, tho and thru, are making not a little quiet
progress. I read a great many manuscripts by American authors, and
find in them an increasing use of both forms, with the occasional
addition of altho, thoro and thoroly. The spirit of American spelling is
on their side. They promise to come in as honor, bark, check, wagon
and story came in many years ago, as tire,[29] esophagus and
theater came in later on, as program, catalog and cyclopedia came
in only yesterday, and as airplane (for aëroplane)[30] is coming in
today. A constant tendency toward logic and simplicity is visible; if
the spelling of English and American does not grow farther and
farther apart it is only because American drags English along. There
is incessant experimentalization. New forms appear, are tested, and
then either gain general acceptance or disappear. One such, now
struggling for recognition, is alright, a compound of all and right,
made by analogy with already and almost. I find it in American
manuscripts every day, and it not infrequently gets into print.[31] So
far no dictionary supports it, but [Pg264] it has already migrated to
England.[32] Meanwhile, one often encounters, in American
advertising matter, such experimental forms as burlesk, foto,
fonograph, kandy, kar, holsum, kumfort and Q-room, not to mention
sulfur. Segar has been more or less in use for half a century, and at
one time it threatened to displace cigar. At least one American
professor of English predicts that such forms will eventually prevail.
Even fosfate and fotograph, he says, are bound to be the spellings
of the future.[33]
§ 6](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-80-320.jpg)
![Minor Differences
—Various minor differences remain to be noticed. One is a
divergence in orthography due to differences in pronunciation.
Specialty, aluminum and alarm offer examples. In English they are
speciality, aluminium and alarum, though alarm is also an alternative
form. Specialty, in America, is always accented on the first syllable;
speciality, in England, on the third. The result is two distinct words,
though their meaning is identical. How aluminium, in America, lost
its fourth syllable I have been unable to determine, but all American
authorities now make it aluminum and all English authorities stick to
aluminium.
Another difference in usage is revealed in the spelling and
pluralization of foreign words. Such words, when they appear in an
English publication, even a newspaper, almost invariably bear the
correct accents, but in the United States it is almost as invariably the
rule to omit these accents, save in publications of considerable
pretensions. This is notably the case with café crêpe, début,
débutante, portière, levée, éclat, fête, régime, rôle, soirée, protégé,
élite, mêlée, tête-à-tête and répertoire. It is rare to encounter any of
them with its proper accents in an American newspaper; it is rare to
encounter them unaccented in an English [Pg265] newspaper. This
slaughter of the accents, it must be obvious, greatly aids the rapid
naturalization of a newcomer. It loses much of its foreignness at
once, and is thus easier to absorb. Dépôt would have been a long
time working its way into American had it remained dépôt, but
immediately it became plain depot it got in. The process is
constantly going on. I often encounter naïveté without its accents,
and even déshabille, hofbräu, señor and résumé. Cañon was
changed to canyon years ago, and the cases of exposé, divorcée,
schmierkäse, employé and matinée are familiar. At least one
American dignitary of learning, Brander Matthews, has openly
defended and even advocated this clipping of accents. In speaking
of naïf and naïveté, which he welcomes because we have no exact
equivalent for either word, he says: But they will need to shed
their accents and to adapt themselves somehow to the traditions of](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-81-320.jpg)
![our orthography.[34] He goes on: After we have decided that the
foreign word we find knocking at the doors of English [he really
means American, as the context shows] is likely to be useful, we
must fit it for naturalization by insisting that it shall shed its accents,
if it has any; that it shall change its spelling, if this is necessary; that
it shall modify its pronunciation, if this is not easy for us to compass;
and that it shall conform to all our speech-habits, especially in the
formation of the plural.[35]
In this formation of the plural, as elsewhere, English regards the
precedents and American makes new ones. All the English
authorities that I have had access to advocate retaining the foreign
plurals of most of the foreign words in daily use, e. g., sanatoria,
appendices, virtuosi, formulae and libretti. But American usage
favors plurals of native cut, and the Journal of the American Medical
Association goes so far as to approve curriculums and septums.
Banditti, in place of bandits, would seem an affectation in America,
and so would soprani for sopranos [Pg266] and soli for solos.[36] The
last two are common in England. Both English and American labor
under the lack of native plurals for the two everyday titles, Mister
and Missus. In the written speech, and in the more exact forms of
the spoken speech, the French plurals, Messieurs and Mesdames,
are used, but in the ordinary spoken speech, at least in America,
they are avoided by circumlocution. When Messieurs has to be
spoken it is almost invariably pronounced messers, and in the same
way Mesdames becomes mez-dames, with the first syllable rhyming
with sez and the second, which bears the accent, with games. In
place of Mesdames a more natural form, Madames, seems to be
gaining ground in America. Thus, I lately found Dames du Sacré
Coeur translated as Madames of the Sacred Heart in a Catholic
paper of wide circulation,[37] and the form is apparently used by
American members of the community.
In capitalization the English are a good deal more conservative
than we are. They invariably capitalize such terms as Government,
Prime Minister and Society, when used as proper nouns; they
capitalize Press, Pulpit, Bar, etc., almost as often. In America a](https://image.slidesharecdn.com/5279437-250622000808-bd6d7c76/85/Spectral-Structures-And-Topological-Methods-In-Mathematics-Michael-Baake-82-320.jpg)
Spectral Structures And Topological Methods In Mathematics Michael Baake
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- 5. SCR Baake et al. | Egyptienne F | Pantone 116, 287 | RB 30 mm Spectral Structures and Topological Methods in Mathematics Michael Baake Friedrich Götze Werner Hoffmann Editors Spectral Structures and Topological Methods in Mathematics Michael Baake, Friedrich Götze and Werner Hoffmann, Editors This book is a collection of survey articles about spectral structures and the application of topological methods bridging different mathematical disciplines, from pure to applied. The topics are based on work done in the Collaborative Research Centre (SFB) 701. Notable examples are non-crossing partitions, which connect representation theory, braid groups, non-commutative probability as well as spectral distributions of random matrices. The local distributions of such spectra are universal, also representing the local distribution of zeros of L-functions in number theory. An overarching method is the use of zeta functions in the asymptotic counting of sublattices, group representations etc. Further examples connecting probability, analysis, dynamical systems and geometry are generating operators of deterministic or stochastic processes, stochastic differential equations, and fractals, relating them to the local geometry of such spaces and the convergence to stable and semi-stable states. ISBN 978-3-03719-197-2 www.ems-ph.org S e r i e s o f C o n g r e s s R e p o r t s S e r i e s o f C o n g r e s s R e p o r t s Spectral Structures and Topological Methods in Mathematics Michael Baake, Friedrich Götze and Werner Hoffmann, Editors
- 7. EMS Series of Congress Reports EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowroński (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowroński and Kunio Yamagata (eds.) Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.) Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.) Derived Categories in Algebraic Geometry.Toyko 2011, Yujiro Kawamata (ed.) Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and Andrzej Skowroński (eds.) Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.) Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.) Functional Analysis and Operator Theory for Quantum Physics.The Pavel Exner Anniversary Volume, Jaroslav Dittrich, Hynek Kovařík and Ari Laptev (eds.) Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Jarosław Buczyński, Mateusz Michałek and Elisa Postinghel (eds.) Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, Fritz Gesztesy, Harald Hanche-Olsen, Espen R. Jakobsen,Yurii Lyubarskii, Nils Henrik Risebro and Kristian Seip, Editors
- 8. Michael Baake Friedrich Götze Werner Hoffmann Editors Spectral Structures and Topological Methods in Mathematics
- 9. mbaake@math.uni-bielefeld.de goetze@math.uni-bielefeld.de hoffmann@math.uni-bielefeld.de Editors: Michael Baake Friedrich Götze Werner Hoffmann Fakultät für Mathematik Universität Bielefeld Postfach 100131 33501 Bielefeld Germany 2010 Mathematics Subject Classification: Primary 58J65, 52C23, 20F65, 11M41; secondary 46L54, 60J45, 35C07, 35Q55, 43A25, 20F36, 11R42, 14L05. Key words: Universal distributions, free probability, Markov processes, Schrödinger operators, heat kernel, spatial ecology, metastability, numerical analysis, critical regularity, aperiodic order, dynamical systems, special Kähler structure, non-crossing partitions, localising subcategory, braided groups, zeta functions, subgroup growth, representation growth, Brumer–Stark conjecture, p-divisible groups. ISBN 978-3-03719-197-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2019 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich, Switzerland Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TeX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1
- 10. Preface This book is a collection of survey articles on several fields of mathematics in which spectral structures appear and topological methods are applied. Those were the over- arching themes under which a large group of researchers joined their efforts in the Collaborative Research Centre (SFB) 701 over three funding periods from 2005 until 2017. The topics span diverse mathematical disciplines from stochastics and dynam- ical systems via global analysis and representation theory to arithmetic geometry. Each article exposes recent results obtained by members and guests of the SFB 701 and embeds them into the general state of the art in the pertinent field. The interre- lations between seemingly disparate areas are demonstrated by the introduction, by several joint papers and by various cross-references between the individual articles. For example, universal probability distributions (Chapter 1) have conjectural con- nections to the zero distribution of zeta functions in number theory, and the special values of those zeta functions carry arithmetic information (Chapter 16). Zeta func- tions are also used to study the growth of the number of representations of a group depending on the degree (Chapter 15). Noncrossing partitions appear in represen- tation theory (Chapter 11), asymptotic distributions (Chapters 1, 11) and geometric group theory (Chapters 11, 13). Another example are generating operators of stochas- tic processes, which are studied in the framework of stochastic differential equations (Chapter 2), of Markov processes in continuum (Chapter 4) and of processes on more general spaces such as fractals (Chapter 3). Thus, the current volume gives insight into recent developments, and highlights the unity of mathematics. We hope that the joint index helps to enhance the usefulness of this publication. The editors would like to use this occasion to express their gratitude to the nu- merous people who made the SFB run smoothly, including Nadja Epp and Stephan Merkes, who managed the general administration including the visitor programme, the student assistants, who ran the IT infrastructure, and the secretaries of the mem- bers, in particular Anita Lydia Cole, who supported the speaker. Also, we express our gratitude to Britta Heitbreder for her expert L A TEX work in preparing this volume. Last not least, we thank the German Research Foundation (DFG) for having invested in this project and for efficient and supportive procedures. Michael Baake Friedrich Götze Werner Hoffmann
- 12. Introduction “Spectral structures and topological methods in Mathematics” Collaborative Research Centre 701 (2005–2017) Over the 12 years of funding, the CRC covered a broad spectrum of research. Its participants were driven by the vision of reinforcing and building bridges between various branches of theoretical and applied mathematics. Many significant develop- ments in mathematics are related to spectral structures and use topological methods. Frequently, they have their origins in applied fields, for example in new concepts of mathematical physics and fluid dynamics, crystallography and materials science. The research pursued in the framework of the CRC may be attributed to one or several of the following broad mathematical topics: asymptotics and universality lattices representation theory harmonic and geometric methods deterministic and stochastic dynamics moduli spaces p-adic L-functions p-adic cohomology In this volume, we would like to highlight some illustrating examples of this re- search, to embed it into a wider mathematical context, and to emphasise connections within and between these topic areas. Let us now give a synopsis of the various chapters. Asymptotic approximations are a major tool for the analysis of distributions in different areas of mathematics. In Chapter 1 (Götze and Kösters), they are used to investigate the accuracy of universal statistical laws for local and global distribution of spectral values of random matrices and sums with independent parts. Here, uni- versality for large times or for large complexity means that these distributions show an emergent collective behaviour in the limit which is independent of special proper- ties of the model, such as the starting distribution, the details of the dynamics or the details of the distribution of the constituent parts. The asymptotic growth of numbers of geometric or algebraic objects are a com- mon theme of Chapter 9 (Baake, Gähler, Huck and Zeiner) as well as Chapter 15 (Voll). The first one considers the enumeration of particular lattices in Euclidean space, the second one centres around the enumeration of subrings and representa- tions of unipotent group schemes. In both cases, these numbers are studied via an analytic encoding in zeta functions, which generalise those of Hecke and Tate.
- 13. viii Other generalisations of these zeta functions with applications to Arthur–Selberg trace formulas in the framework of the Langlands program are reviewed in Chapter 14 by Hoffmann. Their analytic continuation and functional equations are obtained by tools of harmonic analysis like Poisson summation. Apart from enumeration, Chapter 9 (Baake, Gähler, Huck and Zeiner) also fo- cuses on harmonic properties of lattices and quasiperiodic sets as well as on spectral implications of quasiperiodic tilings, their generation and connection to dynamical systems. Similarly, spectral properties of nonlinear dispersive equations of Schrödinger type, studied in Chapter 7 (Herr), are closely tied to spectral sets of lattice points which are investigated by methods of harmonic analysis parallel to those used in an- alytic number theory and the geometry of numbers studied in Chapter 1 (Götze and Kösters). Stochastic dynamical systems and their spectral properties represent another main topic of the CRC 701. In Chapter 5, Gentz studies metastability in parabolic SPDEs and other noise-induced phenomena in coupled dynamics by means of harmonic and stochastic analysis, large-deviation methods and random Poincaré maps. Equivariant dynamical systems are investigated in Chapter 6 by Beyn and Otten, where the surprising stability of equivariant evolution equations and their relative equilibra is studied under numerical discretisation. The stability of waves can be analysed using holomorphic nonlinear eigenvalue problems. In Chapter 8, Kassmann applies methods from partial differential equations and nonlocal operators in Euclidean spaces to study variational solutions of fractional Dirichlet problems and related Harnack inequalities. In Chapter 4, Kondratiev, Kutovyi and Tkachov study Markov birth and death processes in spatial ecologies by means of evolutions of configuration sets and semi- group methods, passing from microscopic stochastic configuration processes to meso- scopic kinematic model equations. In Chapter 2, Röckner discusses open problems and new approaches in solving Fokker–Planck–Kolmogorov equations in finite and in particular in infinite-dimen- sional spaces. He also reviews important results on the corresponding stochastic differential equations in this general infinite-dimensional setup and discusses appli- cations to stochastic differential equations such as the stochastic porous media equa- tion. A panorama of views showing the interplay of different fields of mathematics is developed for the notion of a poset (or lattice) of non-crossing partitions in Chap- ter 11 (Baumeister, Bux, Götze, Kielak and Krause). Non-crossing partitions are fundamental in moment computations for universal laws of non-commutative con- volutions in free probability, and the Kreweras complement in this poset allows to analyse multiplicative non-commutative convolutions. Similarly, it can be used to determine the classifying spaces of braid groups and to describe the Hurwitz action in finite Coxeter systems. Last but not least, non-crossing partitions lie at the heart of bijections between subcategories of thick and coreflective subcategories related to crystallographic Coxeter systems in representation theory. These fruitful connections have been the topic of several conferences hosted by the CRC 701 in the last years on
- 14. ix free probability, quantum groups, algebraic combinatorics, buildings and representa- tion theory. Connections between the algebraic geometry of the projective line and its co- homological localisations in representation theory are reviewed in Chapter 12 by Krause and Stevenson. The example of the projective line shows that the classifi- cation of thick subcategories via non-crossing partitions that arises in representation theory is nicely complemented by the classification of thick tensor ideals arising in algebraic geometry. Harmonic analysis and stochastic dynamics on Riemanian manifolds together with associate heat kernel bounds and escape rates are studied in Chapter 3 (Grigor’yan). Some of these results can be transferred to (ultra-)metric measure spaces by means of Dirichlet forms. For discrete spaces like graphs, the notions of Hochschild homology and other fundamental homology constructions like Künneth’s formula can be partially extendend. In a similar vein, complexes for graphs and surfaces are studied in Chapter 13 (Bux) together with Morse functions on cell complexes in order to access higher finiteness properties of braided version of certain groups. Chapter 10 (Callies, Haydys) is devoted to the interplay of local and global geom- etry and harmonic analysis for special affine Kähler structures. Finally, p-adic analysis, number theory and geometry are in the focus of Chap- ter 16 (Nickel), reporting evidence for conjectures by Gross and Stark on vanishing orders and leading terms of p-adic L-functions and complex L-functions at zero. The theory of displays and the classification of p-divisible groups and with its important recent applications are studied in Chapter 17 by Zink. The research within the CRC 701 established viable connections at the interface between theoretical and applied mathematics: Algebraic geometry and dynamical systems, representation theory and probability theory, stochastic analysis and numer- ics, harmonic analysis connecting nonlinear partial differential equations, stochastics and analytic number theory. The special added value of the CRC 701 has been to realise the full potential of the mathematical theories around these interfaces, which motivated the newly recruited researchers to engage themselves into a coherent, stim- ulating research environment. Establishing such a framework of interactions between different fields of mathe- matics might be viewed as the most important legacy of the CRC 701. Friedrich Götze Michael Röckner Thomas Zink
- 16. Contents 1 Convergence and asymptotic approximations to universal distributions in probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 by F. Götze, H. Kösters 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Symmetric random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Non-symmetric random matrices . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Local spectral distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Connections between probability theory and number theory . . . . . 16 1.6 Analogies between classical and free probability . . . . . . . . . . . . . 20 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Kolmogorov operators and SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 29 by M. Röckner 2.1 Fokker–Planck–Kolmogorov equations . . . . . . . . . . . . . . . . . . . 29 2.2 Three selected results on SPDEs . . . . . . . . . . . . . . . . . . . . . . . . 40 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 Analysis and stochastic processes on metric measure spaces . . . . . . . . 55 by A. Grigor’yan 3.1 Analysis on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Analysis on metric measure spaces . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Homology theory on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 Markov evolutions in spatial ecology: From microscopic dynamics to kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 by Yu. Kondratiev, O. Kutovyi, P. Tkachov 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Complex systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Markov evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Birth-and-death evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.5 Vlasov-type scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
- 17. xii Contents 5 Metastability in randomly perturbed dynamical systems: Beyond large- deviation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 by B. Gentz 5.1 Large deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Kramers’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.3 Parabolic SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4 Unstable periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5 Mixed-mode oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6 Computation and stability of waves in equivariant evolution equations 129 by W.-J. Beyn, D. Otten 6.1 Equivariant evolution equations . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 The freezing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.4 Relative Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.5 Nonlinear eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . 152 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7 Initial value problems for nonlinear dispersive equations at critical regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 by S. Herr 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.2 Nonlinear Schrödinger equations on compact manifolds . . . . . . . . 165 7.3 Nonlinear systems on Euclidean space . . . . . . . . . . . . . . . . . . . . 173 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8 Variational solutions to nonlocal problems . . . . . . . . . . . . . . . . . . . . 183 by M. Kaßmann 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.2 Variational solutions to the Dirichlet problem . . . . . . . . . . . . . . . 185 8.3 Ellipticity and coercivity of nonlocal operators . . . . . . . . . . . . . . 189 8.4 (Weak) Harnack inequalities, and Hölder regularity . . . . . . . . . . . 191 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 9 Spectral and arithmetic structures in aperiodic order . . . . . . . . . . . . 197 by M. Baake, F. Gähler, C. Huck, P. Zeiner 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.2 Weak model sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 9.3 A decorated quasiperiodic tiling with mixed spectrum . . . . . . . . . 203 9.4 Random inflations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 9.5 Enumeration of lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
- 18. Contents xiii 10 Affine special Kähler structures in real dimension two . . . . . . . . . . . . 221 by M. Callies, A. Haydys 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.2 Special Kähler geometry in local coordinates . . . . . . . . . . . . . . . 223 10.3 Some global aspects of special Kähler geometry on P1 . . . . . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11 Non-crossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 by B. Baumeister, K.-U. Bux, F. Götze, D. Kielak, H. Krause 11.1 The poset of non-crossing partitions . . . . . . . . . . . . . . . . . . . . . 235 11.2 Non-crossing partitions in free probability . . . . . . . . . . . . . . . . . 240 11.3 Braid groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 11.4 Non-crossing partitions in Coxeter groups . . . . . . . . . . . . . . . . . 259 11.5 The Hurwitz action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.6 Non-crossing partitions arising in representation theory . . . . . . . . 267 11.7 Generalised Cartan lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 11.8 Braid group actions on exceptional sequences . . . . . . . . . . . . . . . 270 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 12 The derived category of the projective line . . . . . . . . . . . . . . . . . . . . 275 by H. Krause, G. Stevenson 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 12.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 12.3 Types of localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 12.4 Cohomological localisations for the projective line . . . . . . . . . . . 285 12.5 Exotic localisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 13 Higher finiteness properties of braided groups . . . . . . . . . . . . . . . . . 299 by K.-U. Bux 13.1 Introduction: From group theory to topology . . . . . . . . . . . . . . . 299 13.2 Brown’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 13.3 Combinatorial Morse theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 13.4 Matching complexes for graphs and surfaces . . . . . . . . . . . . . . . 305 13.5 Higher generation in symmetric groups and braid groups . . . . . . . 308 13.6 The braided Thompson group V br . . . . . . . . . . . . . . . . . . . . . . . 310 13.7 A cube complex for V br . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 13.8 The Morse function and its descending links . . . . . . . . . . . . . . . . 314 13.9 Connectivity of descending links . . . . . . . . . . . . . . . . . . . . . . . . 316 13.10 Finiteness properties of V br : Proof of Theorem 13.6.1 . . . . . . . . . 320 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
- 19. xiv Contents 14 Zeta functions and the trace formula . . . . . . . . . . . . . . . . . . . . . . . . 323 by W. Hoffmann 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 14.2 Zeta integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 14.3 The Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 14.4 Unipotent terms in the trace formula . . . . . . . . . . . . . . . . . . . . . 338 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 15 Zeta functions of groups and rings—functional equations and analytic uniformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 by C. Voll 15.1 Zeta functions associated to groups and rings . . . . . . . . . . . . . . . 345 15.2 Submodule zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 15.3 Representation zeta functions for unipotent group schemes . . . . . . 355 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 16 Conjectures of Brumer, Gross and Stark . . . . . . . . . . . . . . . . . . . . . 365 by A. Nickel 16.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 16.2 The abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 16.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 16.4 Relations to further conjectures and results . . . . . . . . . . . . . . . . . 379 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 17 Displays and p-divisible groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 by T. Zink 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 17.2 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 17.3 Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 17.4 Classification of p-divisible groups . . . . . . . . . . . . . . . . . . . . . . 397 17.5 The nilpotency condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 17.6 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
- 20. Chapter 1 Convergence and asymptotic approximations to universal distributions in probability F. Götze and H. Kösters The limiting distributions of functionals depending on a large number of independent random variables of comparable size are often universal, leading to a vast number of convergence and approximation results. We discuss some general principles that have emerged in recent years. Examples include classical and entropic central limit theo- rems in classical and free probability, distributions of zeros of random polynomials of high degree and related distributions of algebraic numbers, as well as global and local universality results for spectral distributions of random matrices.1 1.1 Introduction In random matrix theory, the distributions of the eigenvalues of various kinds of ran- dom matrices are investigated, see e.g. [3]. More precisely, we are interested in the asymptotic behaviour of the eigenvalues as the matrix size tends to infinity. Let us begin with one of the most famous results in random matrix theory, the semi-circle law. For each n 2 N, let Xn D .n 1=2 Xjk/16j;k6n be a symmetric Wigner matrix, i.e. a symmetric n n matrix such that .Xjk/16j 6k6n is a family of independent and identically distributed (i.i.d.) real random variables satisfying the moment conditions EXjk D 0 and EX2 jk D 1 (1.1.1) and, for some ı 0, sup n2N sup 16j;k6n EjXjkj2Cı 1 : (1.1.2) The random variables Xjk may depend on n, but this dependence is suppressed in the notation. Let us mention that all the results described in this section continue to hold for symmetric random matrices with non-identically distributed entries, and also for Hermitian Wigner matrices. Let 1; : : : ; n denote the eigenvalues of Xn, and write Xn WD 1 n Pn j D1 ıj for the (empirical) spectral distribution of Xn. The investigation of the limiting spectral 1Projects A4, B1
- 21. 2 F. Götze, H. Kösters distribution (after appropriate rescaling) has a long history and dates back to work by Wigner in the 1950s. The famous semi-circle law states that lim n!1 EXn D sc.dx/ WD 1 2 q .4 x2/Cdx (1.1.3) in the sense of weak convergence. Similar results hold without the expectation, with weak convergence in probability or almost surely. It is worth emphasising that the limiting spectral distribution is always given by the semi-circle distribution sc, ir- respective of the distribution of the random variables Xjk. In this respect, the semi- circle distribution is universal. There are two main approaches to prove the semi-circle law, the method of mo- ments and the method of Stieltjes transforms. If, for each m 2 N, the moments EjXjkjm are finite and uniformly bounded in n, one can use the method of moments. Here one shows using combinatorial arguments that lim n!1 Z xm .EXn /.dx/ D lim n!1 E 1 n tr Xm n D Z xm sc.dx/ (1.1.4) for any m 2 N0. For m D 2l even, the limit is the lth Catalan number, or the number of non-crossing pair partitions of 2l elements. Since convergence in moments implies convergence in distribution when the limiting distribution is determined by its moments, this proves (1.1.3). For any probability measure on the real line, the Stieltjes transform is the ana- lytic function on the upper half-plane CC WD fu C iv 2 C W u; v 2 R; v 0g defined by m.z/ WD Z 1 x z .dx/ ; z 2 CC : (1.1.5) We shall write mXn instead of mXn and msc instead of msc . It is well known that the pointwise convergence of Stieltjes transforms (to a limit which is itself the Stieltjes transform of a probability measure) is equivalent to the weak convergence of the underlying probability measures. In the context of Wigner matrices, the usefulness of the Stieltjes transform comes from the observation that, for each z 2 CC , mXn .z/ D 1 n tr Rn D 1 n n X j D1 1 n 1=2Xjj z n 1 P k;l¤j Xjk Xjl R.j / kl D 1 n n X j D1 1 z 1 n tr Rn C En D 1 z mXn .z/ C En : (1.1.6) Here, Rn WD .Xn zIn/ 1 D .Rkl/ and R.j / n WD .X.j / n zIn 1/ 1 D .R.j / kl / denote the resolvents of the matrices Xn and X.j / n , where X.j / n is obtained from Xn
- 22. Convergence and Asymptotic Approximations 3 by deleting the jth row and the jth column, and En is an error term (see below). If this error term were zero, (1.1.6) would yield the self-consistency equation m2 .z/ C zm.z/ C 1 D 0 ; (1.1.7) whose solution is given by the Stieltjes transform of the semi-circle distribution. Hence, to prove the semi-circle law, one must show that En ! 0 as n ! 1 (in the appropriate sense). More precisely, it is not hard to see that the error term in (1.1.6) is given by .z C mXn .z//En D 1 n n X j D1 Rjj n 1=2 Xjj 1 n X k;l¤j Wk¤l XjkXjlR .j / kl 1 n X k¤j .X2 jk 1/R .j / kk C 1 n tr.Rn R.j / n / ; (1.1.8) so it remains to show that the sums on the right-hand side in (1.1.8) tend to zero as n ! 1. To this end, results from classical probability theory concerning the moments and distributions of (possibly random) linear and quadratic forms of inde- pendent random variables are useful. The problem of optimal approximations of distributions of linear and nonlinear functions of independent random variables is a classical problem of probability and harmonic analysis, which originated from classical analytic number theory. For linear functionals, the arithmetic structure of the summands strongly influences the distri- bution of the sums, as in the Littlewood–Offord problem [96] and in the central limit theorem (CLT). For nonlinear functionals like quadratic forms, this influence is al- ready reduced and studied intensively in the value distribution on lattices by Landau (1924) and the multivariate CLT for balls by Esséen (1945). Hence, one expects that distributions of generic highly non-linear functionals of vectors and matrices of independent variables (e.g. eigenvalues of matrices) exhibit a smooth distributional behaviour irrespective of a possible arithmetic/lattice structure in the distribution of these variables. This expected behaviour applies as well to the distribution of roots of high-degree polynomials with independent random coefficients, which is related to the distribution of algebraic numbers of growing height. Random matrix theory is also related to free probability [3, 102]. To explain this, let us start from the observation that, by generalisation of Eq. (1.1.4), the calculation of the expected traces of more general products of random matrices is also of interest. If .Xn/ and .Y n/ are independent sequences of Wigner matrices as in (1.1.4), it turns out that lim n!1 E 1 n tr X j1 n . lim !1 1 E tr X j1 /In Y k1 n . lim !1 1 E tr Y k1 /In X jm n . lim !1 1 E tr X jm /In Y km n . lim !1 1 E tr Y km /In D 0 (1.1.9) for all m 2 N and all j1; : : :; jm; k1; : : :; km 2 N. By linearity and by induction, this allows us to reduce the expected traces of the products Xj1 n Y k1 n Xjm n Y km n to those
- 23. 4 F. Götze, H. Kösters of the powers Xj n and Y k n. The relation (1.1.9) is called the asymptotic freeness of the sequences .Xn/ and .Y n/. The relation (1.1.9) is the asymptotic counterpart of the notion of freeness in free probability. Here, one considers a unital algebra A endowed with a tracial unital linear functional ' W A ! C and studies the moments '.an / of elements a 2 A. Two elements a; b 2 A are called free (with respect to each other) if ' .aj1 '.aj1 //.bk1 '.bk1 // .ajm '.ajm //.bkm '.bkm // D 0 (1.1.10) for all m 2 N and all j1; : : :; jm; k1; : : :; km 2 N. Just as above, this allows us to reduce the mixed moments '.aj1 bk1 ajm bkm / to the moments '.aj / and '.bk /. The notion of freeness may be regarded as a (non-commutative) analogue of the notion of independence in classical probability theory. Moreover, it allows for the development of a free probability with many parallels to classical probability. For instance, if a and b are free, the moments of the sum aCb and the product ab depend only on the moments of a and b. Thus, provided that all the moment sequences correspond to probability measures a, b etc. on the real line, one may define the free additive and multiplicative convolution of a and b by setting a b WD aCb and a b WD ab : For the multiplicative convolution, we shall assume without further notice that the dis- tributions a and b are supported on the positive half-line. One may then investigate similar questions (limit theorems, infinite divisibility, asymptotic approximations and expansions, . . . ) as for the classical convolutions. For instance, the free CLT shows that if '.a/ D 0 and '.a2 / D 1, then limn!1 n a= p n D sc weakly, which is the direct analogue of the classical CLT. Thus, the semi-circle distribution plays a similar role in free probability as the normal distribution in classical probability. When the probability measures a and b have compact support, one may also take an analytic approach based on suitable transforms, namely Voiculescu’s R- and S-transforms [3, 102], instead of the combinatorial approach outlined above. These transforms are analytic on certain domains in the complex plane and satisfy Rab D Ra C Rb and Sab D Sa Sb ; and hence may be viewed as analogues of the logarithmic Fourier transform and the Mellin transform in classical probability theory. Incidentally, the R- and S-trans- forms are also closely related to the Stieltjes transform in (1.1.5). For instance, for the R-transform, we have Ra .z/ D . ma .z//h 1i z 1 , where . /h 1i denotes the inverse function. In terms of random matrices, the free convolutions may be interpreted as fol- lows: Suppose that .Xn/ and .Y n/ are sequences of self-adjoint random matrices of increasing dimension whose mean spectral distributions converge to X and Y in moments and which are asymptotically free. (Roughly speaking, this means that their
- 24. Convergence and Asymptotic Approximations 5 eigenspaces are in general position to one another.) Then, the limiting mean spectral distributions of Xn C Y n and XnY n are given by X Y and X Y , respec- tively. Thus, the limiting spectral distributions of certain composite random matrices may be investigated using tools from free probability. As already mentioned, the semi-circle distribution may be regarded as the counter- part of the normal distribution in free probability. It seems natural to study functionals under which these distributions have certain extremal properties. For instance, it is well known that the normal distribution maximises Shannon entropy among all distri- butions with mean 0 and variance 1. Similarly, the semi-circle distribution maximises Voiculescu’s free entropy in the same class. Thus, it seems natural to investigate limit theorems with respect to the divergence measures associated with these entropies. This brings us to the field of entropic limit theorems; see Section 1.6.2. In the semi-circle law, in the limit as n ! 1, the eigenvalues are confined to a bounded interval. This is the global (or macroscopic) level, where one studies the weak convergence of the empirical eigenvalue distribution to some limiting distribu- tion (typically with compact support). One may also consider the local (or micro- scopic) level, where the eigenvalues are rescaled in such a way that the mean spacing between neighbouring eigenvalues is of the order 1. One then studies the asymptotic correlations between a small number of eigenvalues which are close to one another. Interestingly, here the limiting distributions are often universal too, which means that the same limits arise for different kinds of random matrices (from the same sym- metry class). See Section 1.4 below for a sample of results. Moreover, these limits also appear in a variety of other contexts. For instance, the asymptotic local spectral distributions for Hermitian Wigner matrices also show up in representation theory (asymptotics of Young diagrams), probability theory (non-colliding stochastic pro- cesses, repulsive particle systems), number theory (zeros of L-functions) and physics (quantisations of chaotic dynamical systems). 1.2 Symmetric random matrices In this section we discuss several extensions of the semi-circle law that have been obtained in the last few years. We continue with the notation and the assumptions for symmetric Wigner matrices from the Introduction. 1.2.1 Rate of convergence in the semi-circle law It is natural to ask for the rate of convergence in the semi-circle law. Given two probability measures and on the real line with distribution functions F and G, we write k k1 WD sup x2R jF.x/ G.x/j for the Kolmogorov distance between and . Then, one may consider the distance either for the mean spectral distribution, n WD kEXn sck1 ;
- 25. 6 F. Götze, H. Kösters or for the spectral distribution, n WD kXn sck1 : The problem to establish upper bounds on n has a long history. In 1993, Bai [7] derived the rate O.n 1=4 / under a 4th moment condition. The rate O.n 1=2 / was obtained independently by Girko (1998, 2002), Bai, Miao and Tsay (2002), and Götze and Tikhomirov (2003) under various moment conditions; see [69] for references. Next, the optimal rate O.n 1 / was obtained for the special case of random matrices with Gaussian entries, first in the Hermitian case [65] and then in the symmetric case [100]. Using concentration of measure techniques, Bobkov, Götze and Tikhomirov [22] obtained the rate O.n 2=3 / under the assumption that the matrix entries satisfy a Poincaré inequality. The optimal rate of convergence O.n 1 / under weak moment conditions was finally established by Götze and Tikhomirov [68, 69], initially under an 8th moment condition [68] and later under a .4 C ı/th moment condition [69]. More precisely, it was shown in [69] that if M4Cı WD sup n2N sup 16j;k6n EjXjkj4Cı 1 (1.2.1) for some ı 0, then there exists a constant C D C.ı; M4Cı / such that n 6 Cn 1 (1.2.2) for all n 2 N. The proof of this result required three major ingredients: 1. A suitable variant of the smoothing inequality to bound the Kolmogorov distance in terms of the difference of the corresponding Stieltjes transforms, see Proposi- tion 2.1 in [68]. This inequality uses a special contour which stays away from the end-points ˙2 of the support of sc, where the Stieltjes transforms are more difficult to control. 2. A recursive argument to derive good bounds on the diagonal entries of the resol- vent close to the real axis, see Section 5 in [68]. Roughly speaking, this argument shows that a bound on EjRjj j2p at distance v from the real axis entails a bound on EjRjj jp at distance v=s0 from the real axis. It was inspired by similar results, albeit under stronger moment conditions, by Cacciapuoti, Maltsev and Schlein [26]. The proof under weak moment conditions is more involved, and uses recur- sive expansions for the resolvent entries similar to (1.1.6), as well as Burkholder’s inequality for martingale difference sequences to estimate the resulting quadratic forms. 3. A recursive inequality for EjmXn .z/ mSC.z/j2 , see Lemma 7.24 in [68]. This inequality is based on a Stein-type expansion adapted to the self-consistency equa- tion (1.1.7), which facilitates the recursion argument considerably. Using moment matching techniques, a simplified proof of (1.2.2) could be given in [61].
- 26. Convergence and Asymptotic Approximations 7 Let us now turn to upper bounds on n. In 1997, Bai, Miao and Tsay [9] ob- tained the rate OP.n 1=4 / under a 4th moment condition. The rate OP.n 1=2 / was established in [64] under a 12th moment condition and in [10] under a 6th mo- ment condition. More recent results by Erdős, Yau and Yin [42] imply that n D OP.n 1 .log n/C log log n /, see e.g. Section 1 in [68]. In [60], complemented by addi- tional material in [58, 59], Götze, Naumov and Tikhomirov proved that, under the condition (1.2.1), n D OP.n 1 log n/ ; (1.2.3) with some explicit constant D .ı/. In view of a result by Gustavsson [72] for GUE matrices (i.e. Hermitian Wigner matrices with Gaussian entries), it seems clear that the optimal rate cannot be better than OP.n 1 log1=2 n/. Thus, a result of the form (1.2.3) is optimal up to logarithmic factors. Götze, Naumov, Tikhomirov and Timushev [61] improved the result (1.2.3) by showing that it is possible to take D 2. More generally, they showed that, un- der the condition (1.2.1), there exist positive constants C D C.ı; M4Cı / and c D c.ı; M4Cı/ such that for 1 6 p 6 c log n, one has P. n K/ 6 Cp log2p n Kpnp for all K 0 : These results for n were obtained by refining the methods developed in [68]. For instance, in [60], the previous estimates from [68] were extended to the off-diagonal entries of the resolvent, and Stein-type expansions were employed systematically in order to bound the pth moment of the error term En in (1.1.8). Moreover, in [61], the authors used moment matching techniques (motivated by results in [37, 83]) to compare a general Wigner matrix to a suitable Wigner matrix with sub-Gaussian entries. Finally, an essential ingredient in all these works was an appropriate version of the local semi-circle law, which will be described in the next subsection. 1.2.2 Local semi-circle law As mentioned below (1.1.5), the proof of the semi- circle law amounts to showing that jmXn .u C iv/ msc.u C iv/j ! 0 as n ! 1, for any fixed u 2 R, v 0. In the last few years, similar results have been obtained for the situation where v may tend to zero as n tends to infinity, but not too fast [40, 38, 60, 61]. More precisely, under suitable moment conditions, jmXn .u C iv/ msc.u C iv/j 6 .log n/C log log n nv (1.2.4) with high probability, uniformly in u 2 R and 1 v log n=n, say. A result of the form (1.2.4) is called a local semi-circle law in the literature, since it may be used as a starting point for the investigation of the local distribution of the eigenvalues. In fact, the local semi-circle law was originally developed by Erdős, Schlein and Yau [40] on their way to proving the universality of the local spectral distribution of
- 27. 8 F. Götze, H. Kösters general Wigner matrices, see also Section 1.4. The first version of the local semi- circle law was derived under the assumption that the underlying matrix entries have finite exponential moments (as well as further regularity properties). This assump- tion was successively relaxed to .4 C ı/th moments in a series of papers by Erdős, Knowles, Schlein, Yau and Yin; compare [38] and the references therein. The paper [60] provided an alternative self-contained proof of the local semi- circle law, also under a .4 C ı/th moment condition, by building upon the techniques developed for the investigation of the rate of convergence [68]. One of the main advantages of this approach is that the exponent of log n in (1.2.4) is reduced from C log log n to a constant (at least in a certain region for the arguments u and v), with a precise dependence on ı. These results were further improved by Götze, Naumov, Tikhomirov and Timu- shev [61], who showed that for any ı 0, there exist constants C0; C1; C2 depending only on ı and M4Cı (see (1.2.1)) such that EjmXn .u C iv/ msc.u C iv/jp 6 C0p nv p for all 1 6 p 6 C1 log n, 1 v C2 log n=n and juj 6 2 C v. By taking p of the order log n and using Markov’s inequality, one re-obtains (1.2.4) for 1 v C2 log n=n and juj 6 2 C v, but with the constant exponent D 1 instead of D C log log n for the logarithmic factor. As already mentioned, this version of the local semi-circle law plays a major role in recent advances on the rate of convergence for n. Further applications include the delocalisation of eigenvectors and the rigidity of eigenvalues; see [42, 60, 61] and the references given there. In view of the results by Gustavsson [72], all these results seem to be optimal up to logarithmic factors. 1.3 Non-symmetric random matrices The spectral distributions of non-symmetric random matrices have also been investi- gated. 1.3.1 Circular law For each n 1, let Xn D .n 1=2 Xjk/16j;k6n be a real Girko– Ginibre matrix, i.e. an n n matrix such that .Xjk/16j;k6n is a family of i.i.d. real random variables satisfying the moment conditions EXjk D 0; EX2 jk D 1 .1 6 j; k 6 n/ and, for some ı 0, sup n2N sup 16j;k6n EjXjkj2Cı 1 : Again, the random variables Xjk are allowed to depend on n. Also, all the results described in this section continue to hold for non-symmetric random matrices with
- 28. Convergence and Asymptotic Approximations 9 non-identically distributed matrix entries, as well as for complex Girko–Ginibre ma- trices. Similarly as above, let 1; : : :; n denote the eigenvalues of Xn, and write Xn for the (empirical) spectral distribution of Xn. Of course, this is in general a proba- bility measure on the complex plane now. The famous circular law states that lim n!1 EXn D circ.dz/ WD 1 1 1 1fjzj1gdz .n ! 1/ in the sense of weak convergence, or a similar result without the expectation (with weak convergence in probability or almost surely). Thus, the spectral distribution converges to the uniform distribution on the unit disk in the complex plane, and the limiting distribution is again universal. This result was first stated by Girko in the 1980’s. One of his key ideas was to study the spectral distributions of the matrices Xn by considering the spectral distributions of the Hermitian random matrices .Xn zIn/.Xn zIn/ , for all z 2 C. More precisely, for a probability measure with compact support on the complex plane, let U.z/ WD Z C log j zj.d/ .z 2 C/ (1.3.1) denote the associated logarithmic potential. We shall write UXn instead of UXn . Then, it is easy to see that UXn .z/ D Z 1 0 log.t/.Xn zIn/.Xn zIn/ .dt/ .z 2 C/ ; (1.3.2) and the proof of the circular law boils down to proving the convergence of the corre- sponding logarithmic potentials. It is worth emphasising that the logarithmic potential UXn .z/ is given by an inte- gral over an unbounded function. Thus, to prove the convergence of (1.3.2), say, it is not sufficient to establish the weak convergence of the singular value distributions of the matrices X zIn, but one must additionally control the large and small singular values of these matrices. The real problem is that of controlling the small singular values. In the case where the matrix entries have a smooth density, this problem was solved by Bai [8]. For general distributions, however, this problem remained open. Its solution became possible only recently due to work by Rudelson [91] and Rudelson and Vershynin [92], who derived stochastic lower bounds on the smallest singular value of a non-symmetric random matrix with independent entries. These bounds were obtained by a combination of geometric and probabilistic methods, inter alia concentration inequalities for linear forms of independent random variables. Inde- pendently of each other, Götze and Tikhomirov [66] and Tao and Vu [97] extended these bounds to ‘shifted’ matrices Xn zIn and applied them to complete the proof of the circular law.
- 29. 10 F. Götze, H. Kösters To continue, optimal bounds for the concentration in small balls of weighted sums Pn kD1 Xk ak of vectors ak with i.i.d. real weights Xk crucially depend on the arith- metic properties of the vectors ak if Xk are Rademacher variables taking values ˙1. Arithmetic consequences of strong concentration have been investigated by Nguyen, Tao and Vu (2009, 2011) e.g. in [96, 87]. They called this inverse Littlewood–Offord phenomenon. Eliseeva, Götze and Zaitsev [35, 36] showed that this phenomenon can be de- rived from seminal results of Arak (1980, 1981) and Arak and Zaitsev (1988), see [4] and the references therein, who investigated a similar phenomenon in a more general context. They studied how a large local concentration of the sum Sa WD Pn kD1 Xk ak of i.i.d. variables Xk implies a good approximation of the distribution of Sa by an induced measure of distributions supported on a higher dimensional lattice under a linear map. As shown in [35, 36], this yields information about the arithmetic struc- ture of the sequence ak extending the results of [96, 87]. 1.3.2 Elliptic law After the proof of the circular law, various extensions were con- sidered, see also the survey paper by Bordenave and Chafaï [23]. One line of research was the proof of the elliptic law [85, 86, 55]. Here, one con- siders random matrices Xn D .n 1=2 Xjk/16j;k6n with similar moment properties as in the circular law, but for which the random variables Xjk and Xkj with j ¤ k are correlated, with a fixed correlation coefficient % 2 . 1; C1/. It turns out that the asymptotic spectral distribution is given by the uniform distribution on an ellipse, with a shape determined by %. By and large, the proof of the elliptic law resembles that of the circular law. However, due to correlations, several parts of the proof have to be adapted, including the bounds on the small singular values. Note that one may regard the ‘elliptic’ matrices as interpolations between Wigner matrices and Girko–Ginibre matrices, and their limiting distributions as interpolations between the semi-circular distribution and the circular distribution. 1.3.3 Products of random matrices In a different direction, the spectral distribu- tions of products of independent Girko–Ginibre matrices X1;n Xk;n (all of size n n) were investigated, for k fixed and n ! 1. Similarly as for Girko–Ginibre matrices, both the singular value distributions [2] and the eigenvalue distributions [67, 90] were studied. The distribution of the (squared) singular values may be addressed either via the method of moments, which leads to the Fuss–Catalan numbers, or via the method of Stieltjes transforms, which leads to an algebraic equation of degree k C 1 for the Stieltjes transform of the limiting distribution. After that, the derivation of the eigenvalue distribution again uses Girko’s Hermiti- sation method. Here, the small singular values of the ‘shifted’ matrices X1;n Xk;n zIn may be controlled by using the previous results for a single Girko–Ginibre ma- trix Xn as well as Horn’s inequalities for the singular values of product matrices, see e.g. [73, Thm. 3.3.14]. The final result is that, for fixed k, the limiting eigenvalue
- 30. Convergence and Asymptotic Approximations 11 distribution is universal and given by .k/ circ.z/ D 1 kjzj2.k 1/=k 1 1 1fjzj1gdz : (1.3.3) Remarkably, this is the same limiting distribution as that for the kth power of a single Girko–Ginibre matrix. Furthermore, the same result holds for products of independent ‘elliptic’ random matrices [56, 89]. Somewhat surprisingly, for products with k 2 factors, one also obtains the limiting eigenvalue density (1.3.3), irrespective of the correlation % of the underlying factors. Fluctuation results for linear statistics of singular values of products of random matrices have also been studied [57]. 1.3.4 Functions of random matrices Götze, Kösters and Tikhomirov [53] pro- posed a general framework for the investigation of matrix-valued functions F n.X1;n; : : :; Xk;n/ of independent Girko-Ginibre matrices X1;n; : : : ; Xk;n (all of size n n). One of the main ideas was to exploit the principle of universality, thereby partly avoiding the need to analyse the self-consistency equations for the Stieltjes transforms. In a first step, one identifies a set of conditions under which one may establish the universality of the limiting singular value and eigenvalue distributions. The in- vestigation of the singular values relies on the Stieltjes transform and uses classical truncation and interpolation arguments from probability theory, while the study of the eigenvalues is again based on Girko’s Hermitisation method. In a second step, one determines the limiting spectral distributions in a simple special case, namely for functions of Gaussian random matrices. Here one can take advantage of the fact that these matrices are bi-orthogonally invariant (i.e. their dis- tribution is not changed if we pre- or postmultiply them by orthogonal matrices), and hence asymptotically free. Thus, it becomes possible to determine the limiting spec- tral distributions by using the calculus of S- and R-transforms from free probability (see the Introduction) as well as classical results from logarithmic potential theory, at least in simple situations. It turns out that the underlying assumptions may be verified in a number of exam- ples. In particular, many results on products of independent Girko-Ginibre matrices and their inverses may now be derived within a common framework [53]. Further- more, the same approach may also be used for sums of independent random matrices [99, 79], although a more explicit description of the limiting spectral distributions is available only in a few special cases here. For example, for random matrices of the form X1;n.X2;n Xk;n/ 1 , or even sums of independent copies thereof, the limit- ing spectral distributions are closely related to the symmetric stable distributions from free probability [79].
- 31. 12 F. Götze, H. Kösters 1.4 Local spectral distributions Let us turn to the local spectral distributions. For later comparison, we start with the classical results for the Gaussian unitary ensemble (GUE), see e.g. [3]. This is the probability distribution on the space Hn of Hermitian n n matrices with the matrix density const exp 1 2 n tr X2 dX ; (1.4.1) where dX D Q 16j 6k6n dX jk Q 16j k6n dX= jk denotes Lebesgue measure on Hn. GUE matrices are special in that they are unitarily invariant (i.e. their distribution does not change under conjugation by unitary matrices) and of Wigner type (i.e. their entries are independent random variables up to the Hermiticity constraint). Further- more, the joint distribution of the eigenvalues may be calculated explicitly. It is given by the density const Y 16j k6n jxk xj j2 n Y j D1 exp 1 2 nx2 j dx1 : : : dxn ; (1.4.2) which may be rewritten in determinantal form, det Kn.xj ; xk/ j;kD1;:::;n dx1 : : :dxn : (1.4.3) Here, Kn is a certain kernel given by a sum of products of Hermite polynomials. Due to this representation, the asymptotic behaviour of the spectral distribution of a GUE matrix Xn follows from that of the Hermite polynomials, which is well known. For the global spectral distribution of the matrices Xn, one obtains (of course) the density of the semi-circular distribution, %sc.x/ D 1 2 p .4 x2/C. For the local spectral distribution, one must first rescale the eigenvalues so that the mean spacing between neighbouring eigenvalues is of the order 1. In the bulk of the spectrum, i.e. near a point a 2 . 2; C2/ with %sc.a/ 0, the properly rescaled eigenvalues are given by Q j .Xn/ D n%sc.a/.j .Xn/ a/. One then finds that the local correlations are asymptotically given by the determinant of the sine kernel Ksine.x; y/ WD sin .x y/ .x y/ : That is, for any m 2 N and any smooth function f W Rm ! R with compact support, we have lim n!1 E 0 @.n m/Š nŠ X j1;:::;jm f .Q j1 .Xn/; : : :; Q jm .Xn// 1 A D Z Rm f .x1; : : : ; xm/ det Ksine.xj ; xk/ j;kD1;:::;m dx1 dxm ; (1.4.4)
- 32. Convergence and Asymptotic Approximations 13 where the sum inside the expectation is over all m-tuples with pairwise distinct el- ements. Similar results, although with a different rescaling and a different limiting kernel, the Airy kernel, hold at the edge of the spectrum, i.e. near the points a D ˙2. Remarkably, the asymptotic local distributions are universal in the sense that the same limits arise in a variety of other situations, both inside and outside the field of random matrix theory. 1.4.1 Random matrices The same local correlations have been established for several classes of random matrices. First of all, it has been known for some time now that similar results hold if we replace the GUE with a unitarily invariant random matrix ensemble defined by a potential V with suitable smoothness and growth properties. Here the matrix density and the eigenvalue density are given by const exp. n tr V.X// dX and const Y 16j k6n jxk xj j2 n Y j D1 exp. nV.xj //dx1 : : : dxn ; (1.4.5) respectively. Note that in the special case V.x/ D 1 2 x2 , we re-obtain the results for the GUE. A density of the form (1.4.5) is also called an orthogonal polynomial ensemble. The reason is that one can use the orthogonal polynomials associated with the weight e nV.x/ to rewrite the density (1.4.5) in determinantal form, with a kernel Kn;V , similarly as in (1.4.3). Thus, the asymptotic analysis is again reduced to ortho- gonal polynomials, which can now be analysed e.g. using Riemann–Hilbert problems, see e.g. Deift [34] and the references therein. At the global level, one obtains a deterministic limit density %V with compact support. It is worth emphasising that this density is not universal, but depends on the choice of the potential V . At the local level, after replacing %sc with %V in the rescaling of the eigenvalues, one obtains the same result (1.4.4) as for the GUE in the bulk of the spectrum. Moreover, similar universality results (with the sine kernel replaced by the Airy kernel) hold at the edge of the spectrum. Thus, the local correlations are universal within the class of unitarily invariant ensembles. Strong universality results for the kernel Kn;V including rates of convergence and transitions between bulk and edge regions have been obtained in [80]. For more general random matrix ensembles, however, a closed expression for the joint distribution of the eigenvalues as in (1.4.2) or (1.4.5) is not available anymore, and the analysis becomes considerably more complicated. Johansson [74] investigated the local correlations for the so-called deformed Gaussian unitary ensemble and proved the analogue of (1.4.4). A deformed GUE matrix has the form Xn C aY n, with Xn a Hermitian Wigner matrix, Y n an inde- pendent GUE matrix, and a 0 fixed. The investigation of this ensemble starts from two important observations. Firstly, the eigenvalue density may be represented as a mixture of certain determinantal densities. Secondly, the correlation kernels associ- ated with these determinantal densities admit a double contour integral representation which is suitable for asymptotic analysis.
- 33. 14 F. Götze, H. Kösters Götze, Gordin and Levina [47, 46] investigated the local correlations for a partic- ular fixed-trace ensemble, the so-called fixed Hilbert–Schmidt norm ensemble (HSE), and proved the analogue of (1.4.4). For r 0, this ensemble is given by the unique probability measure on the set fX 2 Hn W tr.X2 / D nr2 g which is invariant with respect to conjugation by unitary matrices. The main idea is that the GUE may be represented as a mixture of the HSE, for different values of r, and that this relation may be inverted in order to deduce the results for the HSE from the corresponding re- sults for the GUE. The implementation of this idea is technically involved, however, and requires extending the parameter r to the complex domain. More recently, it has been proved by Erdős, Péché, Ramírez, Schlein and Yau [39] and by Tao and Vu [98] that (1.4.4) continues to hold for general Hermitian Wigner matrices satisfying certain technical conditions. Consequently, the local correlations are universal within the class of Hermitian Wigner ensembles as well. The proof of this result is based on sophisticated comparison arguments. The main idea in [39] is to compare general Wigner matrices to random matrices from a suitable deformed GUE, but with a ! 0 as n ! 1, and to show that the local correlations are asymptotically the same. To this end, one needs the local semi-circle law (and several related results) as an input. We refer to the survey paper [41] for details. 1.4.2 Products of random matrices It is well known that the singular value and eigenvalue configurations of a Ginibre matrix (i.e. a non-Hermitian random matrix with i.i.d. complex Gaussian entries) have spectral densities of determinantal form. More recently, it has been observed by Akemann, Burda, Ipsen, Kieburg, Wei and others that the same is true for products of independent Ginibre matrices; compare [1] and the references therein. Up to rescaling, the joint density of the squared singular values of a product of m independent Ginibre matrices is given by const det.xk 1 j /j;kD1;:::;n det..Dj 1 w/.xk//j;kD1;:::;n ; (1.4.6) where .Dw/.x/ WD xw0 .x/ and w denotes a certain weight function. For m ¤ 1, this is not an orthogonal polynomial ensemble as in (1.4.5) anymore, but it is still a so-called bi-orthogonal ensemble [24, 82, 1]. Thus, it is also possible to rewrite the density in determinantal form and to investigate the local correlations for m fixed and n ! 1, see e.g. [1] for an overview. In the bulk and at the edge of the spectrum, one still obtains the familiar sine and Airy kernel, while at the origin, one obtains a new kernel which generalises the Bessel kernel known from the case m D 1. For the eigenvalues, analogous results can be derived. Moreover, similar results were obtained for product matrices consisting of independent Ginibre matrices, truncated unitary matrices, their inverses, or combinations thereof. The original derivations of these results were often based on ad hoc methods. In- spired by related investigations in [82], Kieburg and Kösters [76, 77] observed that several of these results can be obtained in a unifying framework by means of the spherical transform. This is a multivariate analogue of the Mellin transform which is suitable for the investigation of products of independent bi-unitarily invariant random matrices. Moreover, the spherical transform also leads to a closer description of the
- 34. Convergence and Asymptotic Approximations 15 relation between the squared singular values and the eigenvalues for these matrices [76]. In particular, it turns out that the class of bi-unitarily invariant random matri- ces corresponding to a density of the form (1.4.6) is closed under taking indepen- dent products [77]. This leads to a fairly large class which includes many prominent ensembles from non-Hermitian random matrix theory and which gives rise to new universality problems. 1.4.3 Repulsive particle systems Götze and Venker [70] considered more general probability measures on Rn with a density of the form f .x1; : : :; xn/ D const Y 16j 6k6n h.xj xk/ Y 16j 6n e nQ.xj / dx1 dxn ; (1.4.7) where h is a non-negative function satisfying h.x/ D x2 .1 C o.1// as x ! 0 and Q is a potential with sufficient growth at infinity. Note that, in the special case where h.x/ D x2 , the density f reduces to that of the eigenvalues of a unitarily invariant random matrix, see (1.4.5). For a general function h, however, the density f does not seem to have a natural spectral interpretation anymore. Moreover, the correlation functions are not known to be determinantal. Still, the density may be viewed as describing an interacting particle system, with a repulsive interaction generalising that between eigenvalues of (Hermitian) random matrices. Götze and Venker [70] proved that, under certain technical conditions, the asymp- totic local correlations of the particles are still given by (1.4.4). The proof is based on a comparison between the ensemble Pn;Q;h defined by (1.4.7) and an appropriate eigenvalue ensemble Pn;V as in (1.4.5). In a first step, one uses a fixed point argu- ment to construct a potential V (depending not only on Q but also on h) such that Pn;Q;h and Pn;V have the same limiting distribution at the global level. In the sec- ond step, one shows that the corresponding local correlations are asymptotically the same. Here, one uses the theory of Gaussian processes to express the change of mea- sures from Pn;V to Pn;Q;h in terms of linear eigenvalue statistics, which may then be controlled using concentration of measure inequalities for the probability measure Pn;V . Kriecherbauer and Venker [81] derived similar universality results at the edge of the spectrum, where one encounters the Airy kernel. Universality for the empirical spacing distribution was shown by Schubert and Venker in [93]. There, also a new lo- cal rescaling of eigenvalues using distribution functions was introduced, which yields better rates of convergence than the previously used rescalings. Moreover, Venker [101] proved a similar result as in [70] for the case where h.x/ D jxjˇ .1 C o.1// as x ! 0, with general ˇ 0. In this case, the asymptotic local correlations in the bulk coincide with those of the Gaussian beta ensemble, which is obtained by replacing Q 16j k6n jxk xj j2 with Q 16j k6n jxk xj jˇ in (1.4.2).
- 35. 16 F. Götze, H. Kösters 1.5 Connections between probability theory and number theory 1.5.1 Correlations of zeros of random polynomials It is a long-standing conjec- ture, supported by partial theoretical results [84] as well as by extensive numerical evidence [88], that the local correlations of the zeros of the Riemann zeta function along the critical line are also governed by the sine kernel. Moreover, further similar- ities emerge if one compares the moments and the (auto)correlations of the Riemann zeta function with those of characteristic polynomials of various classes of unitary and Hermitian random matrices (Keating–Snaith [75], Brézin–Hikami [25], Conrey– Farmer–Keating–Rubinstein–Snaith [33], Strahov–Fyodorov [95], Götze–Kösters [52], Shcherbina [94]). Connections to number theory also arise in another direction. Given that the eigenvalues of a random matrix are the zeros of the corresponding characteristic polynomial, it seems natural to ask for the distribution of the zeros for other classes of (random or non-random) polynomials as well. See for instance Götze, Kaliada and Zaporozhets [49] for the distribution of complex algebraic numbers of increasing height. Note that the transformation from the coefficients to the zeros of a polynomial is a highly non-linear operation, just as the transformation from the entries to the eigenvalues of a matrix. Even more, the Jacobian of both transformations is closely related to the squared Vandermonde determinant, Q 16j k6n jzk zj j2 . In view of this observation, the numerous similarities between zeros of random polynomials and eigenvalues of random matrices come perhaps not too unexpected. Recently, Götze, Kaliada and Zaporozhets [50] investigated the correlations of zeros of random polynomials of degree n of with real coefficients with a joint density. Since a real polynomial can have both real and complex zeros, with the complex zeros symmetric with respect to the real axis, it is convenient to consider, for any k; l 2 N0 with k C 2l 6 n, the mixed correlation functions %k;l describing the correlations between k eigenvalues on the real line and l eigenvalues in the upper half-plane. These correlation functions also contain a Vandermonde factor, which shows that adjacent zeros tend to repel one another. In a different direction, Götze, Kaliada and Zaporozhets [51] investigated the cor- relations between conjugate algebraic numbers. More precisely, consider all prime polynomials with integer coefficients of degree n and height bounded by Q, for var- ious choices of height, and count the corresponding .k C l/-tuples of conjugate al- gebraic numbers with values in a given subset of Rk Cl C. Then, it can be shown that, after appropriate normalisation, these numbers converge to a limit as Q ! 1. Furthermore, at least for some choice of the height function, the limiting correla- tion functions coincide with the known correlation functions of the zeros of certain random polynomials. 1.5.2 Diophantine approximations in metric number theory The problem of Dio- phantine inequalities for linear forms on Zd with coefficients taken from a submani- fold of Rd has been studied by Mahler, Khinchine and more recently by Schmidt and Sprindžuk. In the last decade Margulis, Kleinbock, Bernik and Beresnevich solved
- 36. Convergence and Asymptotic Approximations 17 important problems concerning Diophantine inequalities for one-dimensional sub- manifolds given by .x; x2 ; : : : ; xd / 2 Rd ; x 2 R, and more generally for higher- dimensional submanifolds defined by a non-degenerate system of functions. In par- ticular, the breakthrough solution of the Baker–Sprindžuk conjecture due to Klein- bock and Margulis [78] that used the dynamics of flows on homogeneous spaces led to a surge of activity in a range of related long-standing problems. Some advances of more recent years include progress on the distribution of rational and algebraic points near manifolds [12, 15, 48], the distribution of discriminants and resultants of integral polynomials [16] as well as sharp quantitative bounds for the number of pairs of close conjugate algebraic numbers of a fixed degree n and bounded height [13]. More precisely, Beresnevich, Bernik and Götze [13] studied for arbitrary con- jugate algebraic numbers over Q (resp. algebraic integers) ˛1 ¤ ˛2 the universal minimal exponent, say D n (resp. D n ), such that the inequality j˛1 ˛2j H.˛1/ holds for sufficiently large H.˛1/. Here H.˛/ denotes the height of the algebraic number ˛, i.e. the absolute height of the minimal polynomial of ˛ over Z. Clearly n 6 n. Mahler proved in 1964 that n 6 n 1. For n 6 3, satisfac- tory bounds have been shown by Bugeaud, Evertse and Mignotte. An old result by Mignotte for the general case, i.e., n 3 said that n; n n=4. In [13] it was shown that in fact, for n 2, n; n .n C 1/=3 holds, which follows from a counting re- sult for pairs of real algebraic conjugate numbers in certain intervals of given large height. This is in turn a consequence of results in metric number theory which gen- eralises results of Baker, Schmidt, Bernik, Kleinbock and Margulis. For an overview of techniques and results, see [14]. 1.5.3 Effective bounds in Diophantine inequalities A classical problem of effec- tive bounds means to determine the size of the smallest vector m 2 Zd n f0g such that the Diophantine inequality jQŒmj 1 holds. Here, QŒx denotes an (non- degenerate) indefinite quadratic form with real coefficients on Rd . For forms with integer coefficients it is known by a result of Meyer (1884) that there exists an m 2 Zd n f0g with QŒm D 0 provided that d 5. As for the size of this solu- tion vector, classical results of Birch and Davenport (1958) using the geometry of numbers show that for indefinite integral quadratic forms 0 QCŒm cd j det Qj holds, where QC D .QT Q/1=2 is positive definite. For diagonal indefinite quadratic forms Q, Birch and Davenport (1958) have shown that there exists a nontrivial solution m ¤ 0 to jQŒmj of size kmk ı 2Cı in dimension d D 5. In higher dimensions solutions may be generated via embedding. Using theta functions, methods from the geometry of numbers and the asymptotic orbit behaviour of unipotent subgroups of SL2.R/, Götze and Mar- gulis proved in [54] for general indefinite QŒx a bound of order 0 kmk ı kd Cı ; kd D 12; 8:5; 7 for d D 5; 6 and d 7. 1.5.4 Lattice point counting problems The existence of m ¤ 0 with jQŒmj is a consequence of precise results for counting lattice pointsin elliptic as well as hyper- bolic shells defined via a quadratic form QŒx in Rd and Ea;r WD fQŒx 2 Œa; rg for
- 37. 18 F. Götze, H. Kösters Q positive definite and Hr .a; b/ WD fQŒx 2 Œa; b; x 2 rCg (C suitable compact convex body) in the case of Q indefinite as r in Hr .a; b/ and Ea;r tends to infinity. The counting error is measured relative to volume, say jE0;rj, resp. jHr .a; b/j of these compact rescaled regions, say Br , via ˇ ˇ card.Zd Br / jBr j ˇ ˇ = jBrj D O.ır r 2˛ /; r ! 1; (1.5.1) where ˛ 6 1. In case that the optimal exponent ˛ D 1 can be shown, the factor ır may be bounded below or tend to zero depending on the Diophantine properties of the coefficients of Q. Such error bounds have been studied for a long time, starting with Landau (1915, 1924), Jarnik (1928) for special positive definite forms QŒx being diagonal or with integer coefficients, where ˛ D 1 could be shown. Optimal exponents have been shown for d 9 and definite as well as indefinite general forms QŒx first by Bentkus and Götze (1997). For dimension d 5, the optimal exponent was shown in [45] for E0;r and in [54] for Hr .a; b/ such that b a tends to infinity. As for the dependence of Diophantine properties of Q, Davenport and Lewis (1972) conjectured that for positive definite irrational forms the distances of the ordered elements of QŒZd for d 5 converge to 0. This is related to the famous conjecture by Oppenheim (1929) that QŒZd is dense in R for indefinite irrational forms for d 5, which was proved by Margulis (1986) even for d 3. Quantitative versions of these conjectures for irrational forms were shown for d 9 in Bentkus and Götze (1997) as well as for d 5 and Q positive-definite in [45]. In [54], the problem for both cases has been solved for d 5 by means of a unified approach. Lower bounds for the error for E0;r in dimensions d D 2; 3; 4 show that the optimal exponents satisfy ˛ 1=4; 1=2; 1 respectively, multiplied by logarithmic factors like .log r/ˇ ; ˇ 0; d D 2; 3 resp. .log log r/ ; 0; d D 4, whereas lower error bounds for dimensions 5 and higher are just given by .r 2 /. Crucial for the proof of ˛ D 1 and ır ! 0 in (1.5.1) is the investigation of theta sums on the generalised Siegel upper half-plane, that is on matrices iQ C A, where A; Q are real symmetric d d matrices with Q positive definite, given by .iQ C A/ D X m2Zd expŒ mT Q m C imT A m: (1.5.2) The number of lattice points m such that Q.m/ 6 r may be expressed as an integral along a theta function .sQ/ expŒitr=.2s/ on a line s D t C ir 2 ; t 2 R (that is a degenerate horocycle), with given by (1.5.2). The following inequality between theta functions turned out to be an essential step: j.sQ/j2 6 .t/ WD c.Q/rd X m;n expŒ Ht .m; n/; (1.5.3) where Ht denotes the t-dependent quadratic form Ht .m; n/ WD r2 km t Q nk2 C r 2 knk2 (1.5.4)
- 38. Convergence and Asymptotic Approximations 19 of .m; n/ 2 Z2d . Using lattice density bounds for .t/, via the first d successive Minkowski minima Mt;1 6 Mt;2 6 : : : 6 Mt;d of Ht , that is via .t/ 6 c.Q/ rd Mt;1 : : :Mt;d ; (1.5.5) the problem of estimating the lattice point remainder is transferred to the estimation of an integral along .sQ/ together with questions in metric number theory and the geometry of numbers. This approach is closely connected to the study of ergodic properties of unipotent and quasi-geodesic flows in the papers of Eskin, Margulis and Mozes (1998). Indeed, we may write the quadratic form Ht .m; n/ as Ht .m; n/ D k r ut .m; n/k2 ; r WD diag.r2 Id; r 2 Id/; ut .m; n/ WD .m tQn; n/; (1.5.6) where r 2 SL.2d; R/; r 0 denotes a quasi-geodesic flow and ut 2 SL.2d; R/; t 2 R a unipotent flow in SL.2d; R/. 1.5.5 Central limit theorems Let B denote a domain in Rd with 0 2 B and smooth boundary, which is symmetric to 0. Let X1; : : :; Xn denote independent and identi- cally distributed Rd -valued random vectors with EX1 D 0; EkX1k4 1 and iden- tity covariance. A classical problem in probability theory is the question of optimal er- ror bounds in the central limit theorem for the distribution of sums Sn D X1C CXn of random vectors on the system of sets Br D r B; r 0, i.e. the determination of the exponent ˛ in sup r0 ˇ ˇ ˇP ˚ Sn 2 n1=2 Br PfS 2 Br g ˇ ˇ ˇ D O.n ˛ /; (1.5.7) where S denotes a random vector Rd with a standard Gaussian distribution. For non-degenerated ellipsoids B, Esséen (1945) has shown ˛ D d=.d C 1/, which has been extended to uniformly convex bodies with smooth boundary. In the case of random vectors with independent coordinates and coefficient matrices which are di- agonal with respect to a lattice basis, the exponent ˛ D 1 has been shown for d 5 by Bentkus and Götze (1996). Identifying n with r2 , there exists an obvious corre- spondence of error bounds in the CLT, that is in (1.5.7), for ellipsoids symmetric to 0 with bounds for the relative lattice point remainder in (1.5.1). The optimal error order which holds uniformly in the distribution of the random vectors subject to centering, moment and covariance conditions only is given by ˛ D 1 in (1.5.7) for d 9 for quadrics (Bentkus–Götze 1997) and ˛ D k=2 for regions defined by special k-th or- der polynomials, k 3 and large d (Götze 1989). The optimal rate for quadrics and d 9 required new techniques from analytic number theory (Bentkus–Götze 1999). This result has been extended to general U -statistics and quadratic forms of n dimen- sional vectors with independent components (Götze–Tikhomirov 2002) provided that their rank d is at least 12 or larger. For a detailed review of the connections between
- 39. 20 F. Götze, H. Kösters lattice point problems and the CLT and the extensive literature in this field, see the reviews in [44] and [71]. The approximation results for Hr .a; b/ described above were essential as well for proving the above rate of convergence ˛ D 1 for quadrics in the CLT down to dimension 5 in [71]. Here, Götze and Zaitsev proved, for example, an explicit error bound for the elliptic or hyperbolic regions of type B WD fQŒx 6 1g in dimensions d 5, sup r0 j PfQŒSn 6 nr2 g PfQŒS 6 r2 g j 6 cd n 1 j det Qj 1=2 EkQ 1=2 Xk4 : This result concludes a long series of investigations starting with the seminal results by Esséen (1945) mentioned above. It relies on the fact that the characteristic func- tion of quadratic forms Q.Sn/ for sums of independent identically distributed vector summands may be estimated via the average over random matrices A in the char- acteristic functions of bilinear forms hAT1; T2i of two independent sums T1; T2 of Rademacher vectors taking values in the lattice Zd . These characteristic functions in turn may be estimated again using local limit theorems via theta sums as outlined in (1.5.3). Hence, the decisive step of controlling the local fluctuations of the distri- bution of such indefinite quadratic forms could be reduced to the estimates outlined above in order to obtain a full correspondence in terms of dimensions and rates be- tween both areas. In order to investigate distribution functions of quadratic forms, a crucial technical obstacle had been to extend the averaging (in t) of the characteristic described in (1.5.5) from the measure dt (sufficient for narrow shells Hr .a; b/ with bounded intervals Œa; b) to the harmonic measure dt=t on the real line. For lower dimensions, for example d D 3, one cannot expect these optimal rates in view of the correspondence to the lattice point error problem in these dimensions. 1.6 Analogies between classical and free probability In non-commutative probability theory the addition and multiplication of ‘free ran- dom variables’ corresponds to the ‘free’ additive and multiplicative convolution of the corresponding spectral probability measures which are represented by Hermitian operators in certain von Neumann algebras A with a finite normalised faithful trace , see e.g. the survey [102]. A guiding principle for the investigation of free convo- lutions of spectral measures is the analogy to the classical theory of convolutions of probability measures. Within the framework of comparison studies between classical and free probabil- ity, Chistyakov and Götze investigated the free additive convolution with respect to a classification of indecomposable elements and infinite divisibility. The final results now appeared in [29]. They represent an analog of the classical results of infinite di- visibility via the so-called Bercovici–Pata bijection, which yields a correspondence of classical Lévy measures to free Lévy measures. The latter appear in free Khintchine type decompositions via integral representations of reciprocal Stieltjes transforms,
- 40. Convergence and Asymptotic Approximations 21 which are analytic functions from CC to C and hence closely related to the class of Nevanlinna functions. A remarkable difference described for free measure decompo- sitions in [29] as compared to classical ones is that the infinitely divisible measures without indecomposable factors are trivial Dirac measures only. This observation holds not only for the free additive convolution, but also for the free multiplicative convolution on the positive half-line as well as on the unit circle [29]. As for analogs of results of classical parametric statistics, the independence of sample mean and sample covariance for Gaussian random variables has a counterpart in free probability for free semi-circular random variables in von Neumann algebras; see Chistyakov, Götze and Lehner [32]. 1.6.1 Asymptotic approximations in free probability In [28], Chistyakov and Götze investigated the rate of convergence of the distribution function of an n-fold normalised free additive convolution of a spectral measure with itself to that of Wigner’s semi-circle law. Since their result showed a complete analogy to the Berry– Esséen theorem in classical probability, i.e. a rate of order O.n 1=2 / assuming the existence of a third moment of , they investigated higher order approximations for this n-fold free convolution of non-trivial measures. Unlike the classical case there are no arithmetic obstructions to the smoothness of such convolutions: for n larger than a finite threshold (depending on ), the resulting measure admits a density with respect to Lebesgue measure. Thus one would hope that, similarly to Edgeworth ex- pansions in classical probability, moment conditions of order k C 2 suffice to define asymptotic approximations up to an error of order o.n k=2 / involving free cumulants. This is indeed the case for k D 1, and the expansion term involves derivatives of the semi-circle density. Differences to the classical case appear for the expansion term of order k D 2, which cannot be written in terms of derivatives of the semi-circle density anymore: one needs signed measures and a slightly shifted support when the third cumulant does not vanish. These problems are essentially caused by the singu- larity of the semi-circle density at the boundary of the support. Technically the proofs for the expansions are analytically much more involved then in the classical case. For k D 1; 2 one has to determine a subordinating function as a particular solution of a 3rd resp. 5th order equation, which degenerates into the quadratic equation (1.1.7) for the Stieltjes transform of Wigner’s semi-circle law as n ! 1, and study its explicit dependence on n. An alternative way to derive expansions of n 1=2 .X1 C : : : C Xn/ in the free CLT for free identically distributed random variables Xj and related approximation problems for random matrices starts from a universal scheme of expansions for se- quences of symmetric functions, see [62]. This scheme is an umbrella limit theorem for all Gaussian limits and related Edgeworth-type expansions in permutation sym- metric functions of many variables by rotational invariant functions and corrections via polynomials in power sums of degree at least two. Applications to the highly non-linear free convolution were done in [63]. It involved derivatives with respect to j at zero of distribution functions like W C 1X1 C 2X2, where W has semicircle distribution. In the interior of the limiting spectral support the validity of these ex-
- 41. 22 F. Götze, H. Kösters pansions is due to the smoothness of the distribution functions of 1X1 C : : : C nXn of .1; : : : ; n/ 2 Rn in certain domains. 1.6.2 Entropic limit theorems Let us return to the classical central limit theorem. Let X1; X2; X3; : : : be i.i.d. real random variables with mean 0 and variance 1, and let S1; S2; S3; : : : denote the associated partial sums. Let fn denote the density of Sn= p n (if existent), and let ' denote the density of the standard normal distribution. The classical local limit theorem states that kfn 'k1 ! 0 if and only if there exists some n0 2 N such that fn0 is bounded. Since the existence of bounded densities is still a strong condition, it seems natural to look for similar characterisations under weaker conditions. One such result is the entropic central limit theorem by Barron (1985), which states that D.fn j '/ ! 0 if and only if there exists some n0 2 N such that D.fn0 j '/ 1. Here, D.f j '/ WD Z L.f .x/='.x// '.x/dx denotes the relative entropy of f with respect to ', and L.x/ WD x log x. A nice interpretation of this result arises from the observation that D.fn j '/ D H.'/ H.fn/, where H denotes Shannon entropy, and that the standard normal distribution maximises Shannon entropy among all random variables of mean 0 and variance 1. Thus, the system converges to the state of maximal entropy. In this respect, let us also mention the result by Artstein, Ball, Barthe and Naor [5] that the entropy tends to that of the standard normal distribution monotonously, in line with the second law of thermodynamics. Moreover, in [6], these authors proved that the rate of convergence is O.n 1 / if the random variables Xn satisfy the Poincaré inequality. However, for more general probability densities, the question of the rate of convergence remained open. This question was answered completely in a series of papers by Bobkov, Chistyakov and Götze [17, 20] from the last few years. Their analysis combined tools from information theory (such as the entropy convolution inequality) with more clas- sical results on characteristic functions and their asymptotic approximation. In [17], the authors derived Edgeworth-type expansions for the entropic central limit theorem. In particular, if the underlying random variables Xj have finite fourth moments, the rate of convergence is still O.n 1 /, and hence much better than in the classical cen- tral limit theorem. This is related to the fact that the term of order O.n 1=2 / in the classical asymptotic expansion of the density fn is an odd function, and hence van- ishes when taking the entropy integral. Moreover, optimal rates for the case where the random variables Xj have finite fractional moments of order 2 s 4 have also been obtained [17]. These results rest on technically involved approximations in the local limit theorem for sums of i.i.d. random variables in the case of fractional moments. Furthermore, Berry–Esséen bounds in the entropic central limit theorem have also been established [20].
- 42. Convergence and Asymptotic Approximations 23 Similar results were also derived for related notions from information theory, e.g. Fisher information [18], as well as for other limit laws, e.g. stable distributions [19]. However, in the latter situation some complications arise. For instance, since higher moments do not exist, asymptotic expansions are not available anymore. Moreover, the full analogue of the entropic limit theorem can only be obtained for the non- extremal stable laws. For the extremal stable laws, additional technical conditions are needed [19]. This might be related to the fact that (non-Gaussian) stable distributions do not maximise the entropy functional anymore. Chistyakov and Götze [31] studied related questions for the entropic free central limit theorem, i.e. they studied the convergence of Voiculescu’s free entropy to its maximum value (under fixed variance) assumed for the semi-circle distribution. As- suming a finite moment of order four, they showed that the rate of convergence in the entropic free CLT is also of the order O.n 1 /. Furthermore, they obtained an ex- pansion up to an error of order o.n 1 /. These results were based on previous results about expansions for densities in the free CLT [30]. Besides sums of i.i.d. random variables, maxima of sums of i.i.d. random variables have also been considered (Bobkov–Chistyakov–Kösters [21]). Here, the limiting distribution is given by the one-sided normal distribution, which also maximises a suitably defined entropy functional. Furthermore, one also obtains a characterisation result here: The maxima of the sums converge to the one-sided normal distribution in relative entropy if and only if the original random variables Xj have finite relative entropy with respect to the one-sided normal distribution. The proof is also based on a combination of the entropy convolution inequality with more classical results for characteristic functions, including Spitzer’s formula. References [1] G. Akemann and J. Ipsen, Recent exact and asymptotic results for products of independent random matrices, Acta Phys. Pol. B 46 (2015), 1747. [2] N. Alexeev, F. Götze and A. Tikhomirov, Asymptotic distribution of singular values of pow- ers of random matrices. Lith. Math. J. 50 (2010), 121–132. [3] G.W. Anderson, A. Guionnet and O. Zeitouni, An Introduction to Random Matrices, Cam- bridge University Press, Cambridge (2010). [4] T.V. Arak and A.Yu. Zaı̆tsev, Uniform limit theorems for sums of independent random vari- ables, Proc. Steklov Inst. Math. 174 (1988), 1–222. [5] S. Artstein, K.M. Ball, F. Barthe and A. Naor, Solution of Shannon’s problem on the mono- tonicity of entropy, J. Amer. Math. Soc. 17 (2004), 975–982. [6] S. Artstein, K.M. Ball, F. Barthe and A. Naor, On the rate of convergence in the entropic central limit theorem, Probab. Th. Rel. Fields 129 (2004), 381–390. [7] Z.D. Bai, Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices, Ann. Probab. 21 (1993), 625–648. [8] Z.D. Bai, Circular law, Ann. Probab. 25 (1997), 494–529.
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- 48. Chapter 2 Kolmogorov operators and SPDEs M. Röckner The purpose of this paper is to survey a number of selected results on Kolmogorov operators and SPDEs. It consists of two parts: One part is related to Kolmogorov operators and is devoted to the corresponding linear Fokker–Planck–Kolmogorov equations (see [11]), the other part is about three key results about SPDEs obtained resp. published during the last funding period, namely an existence and uniqueness result for L2 -initial data for the stochastic total variation flow, a new approach to SPDEs and a pathwise uniqueness result of SDEs on Hilbert spaces with a merely bounded drift part. Both parts are survey type papers coauthored by regular visitors of the CRC 701. The reader who is interested in a more comprehensive and less se- lective overview of the results of Project B4 and also Project A9 is referred to the monographs [1, 5, 11, 19] and the references therein.1 2.1 On recent progress and open problems in the study of linear Fokker–Planck–Kolmogorov equations V.I. Bogachev, M. Röckner and S.V. Shaposhnikov This part reports on some recent progress and remaining challenging open prob- lems in the study of linear elliptic and parabolic Fokker–Planck–Kolmogorov equa- tions obtained in Project B4. 2.1.1 Introduction Let us define (linear) Fokker–Planck–Kolmogorov equations in the elliptic and parabolic cases and formulate several problems related to these equa- tions. Consider the Kolmogorov operator Lu.x/ D d X i;j D1 aij .x/@xi @xj u.x/ C d X iD1 bi .x/@xi u.x/; where aij and bi are Borel functions on Rd such that A D .aij /16i;j 6d is a non- negative symmetric matrix. 1Projects A9, B4
- 49. 30 M. Röckner We say that a bounded Borel measure on Rd (possibly signed) satisfies the Fokker–Planck–Kolmogorov equation L D 0 (2.1.1) if aij ; bi 2 L1 loc.jj/, where jj denotes the variation of , and for every function u 2 C1 0 .Rd / Z Rd Lu d D 0: If is given by a density % with respect to the Lebesgue measure, Eq. (2.1.1) can be written as an equation for the density, d X i;j D1 @xi @xj .aij %/ d X iD1 @xi .bi %/ D 0: (2.1.2) A parabolic Fokker–Planck–Kolmogorov equation is introduced similarly. Consider T 0, and let Lu.x; t/ D d X i;j D1 aij .x; t/@xi @xj u.x; t/ C d X iD1 bi .x; t/@xi u.x; t/; where aij and bi are Borel functions on Rd .0; T / such that A D .aij /16i;j 6d is a non-negative symmetric matrix. We say that a bounded (i.e. of bounded variation) Borel measure on Rd .0; T / (possibly signed) is defined by a family of Borel locally bounded measures .t /0tT on Rd if for every bounded Borel set B the mapping t 7! t .B/ is measurable and .dx dt/ D t .dx/ dt. Below, we will deal with measures of bounded variation. A Borel measure on Rd .0; T / defined by a family of measures .t /0tT on Rd satisfies the Fokker–Planck–Kolmogorov equation @t D L (2.1.3) if aij ; bi 2 L1 loc.jj; Rd .0; T // and if, for every function u 2 C1 0 .Rd .0; T //, Z T 0 Z Rd @t u.x; t/ C Lu.x; t/ dt dt D 0: If is given by a density % with respect to the Lebesgue measure on Rd .0; T /, Eq. (2.1.3) can be written as an equation for the density, @t % D d X i;j D1 @xi @xj .aij %/ d X iD1 @xi .bi %/:
- 50. Kolmogorov operators and SPDEs 31 Let be a locally bounded Borel measure on Rd . We say that a Borel lo- cally bounded measure defined by a family of Borel locally bounded measures .t /0tT satisfies the initial condition ˇ ˇ tD0 D if for every 2 C1 0 .Rd / lim t!0 Z Rd .x/ dt D Z Rd .x/ d: Thus, we study the uniqueness problem for solutions to the Cauchy problem @t D L ; ˇ ˇ tD0 D ; (2.1.4) and we are interested in the two classes of solutions: probability and integrable. A probability solution is a solution defined by a family of probability measures .t /0tT , i.e., t 0 and t .Rd / D 1. The set of all probability solutions D t .dx/ dt such that jbj 2 L2 .; U Œ0; T / for every ball U Rd is denoted by P, where j:j denotes the Euclidean norm on Rd . Among many important and interesting problems connected with Fokker–Planck– Kolmogorov equations and studied in the papers included in the bibliography, we have chosen for discussion here the following two, related by some similarity in the nature of assumptions about the equation coefficients: uniqueness of solutions and bounds on solution densities. Moreover, we confine ourselves to nondegenerate dif- fusion matrices and, what is especially important, to linear equations. Although we have also obtained some results on nonlinear equations and degenerate equations, these directions belong to our future plans. Before addressing the aforementioned main topics, we briefly recall some basic facts related to the existence of solution densities and their properties such as local boundedness and continuity. It was shown in [10] that, in case of non-negative solutions, the nondegeneracy of the diffusion matrix at every point is sufficient for the absolute continuity. It is still an open question whether this is true for signed solutions. For non-negative solutions, there is a more precise result, which we formulate in the elliptic case: the measure .detA/1=d has a density % with respect to Lebesgue measure and % 2 L d=.d 1/ loc ./. Therefore, if 1= detA is locally bounded, the measure itself has a density of this class. This result is sharp in the sense that one cannot omit the factor .det A/1=d in front of and also it is impossible to increase the guaranteed order of integrability d=.d 1/ without additional conditions. For signed solutions, there is the following result. If the functions aij belong to the class VMO (defined below, see also [11]) and on every ball one has 1 Id 6 A.x/ 6 2 Id, where 1; 2 0 are constant, and jbj 2 Lq loc.Rd /, where q d, or, alternatively, jbj 2 L q loc.jj/, where q d, then has a density % 2 Lr loc.Rd / for all numbers r 2 Œ1; C1/. However, these conditions do not ensure local boundedness of solution densities. Moreover, even the continuity of aij and bi along with the nondegeneracy of A is not enough for that. A sufficient condition on A to ensure the continuity of densities is stronger: one needs Dini continuity, i.e., Z 1 0 !.t/ t dt 1
- 51. 32 M. Röckner for the modulus of continuity ! of A. The result is this. Suppose that, on every ball, A has a modulus of continuity satisfying the Dini condition. Let det A 0 everywhere and jbj 2 L q loc.Rd /, where q d. Then, the density % of every solution (possibly, signed) to the elliptic equation L D 0 has a continuous version. If, in addition, the matrix A is locally Hölder continuous of order ı 2 .0; 1/, then % has a density that is locally Hölder continuous of order ı. Actually, the afore- mentioned results are proved for equations on domains, but we formulate them for simplicity in the case of the whole space. Increasing the regularity of A, one gets more properties for solution densities. If the matrix elements aij belong to the local Sobolev class W p;1 loc with p d (which by the Sobolev embedding theorem yields the existence of a continuous version) and A is nondegenerate at every point, while the coefficients bi are either in Lp loc.Rd / or in Lp loc.jj/, then the solution density belongs to the same local Sobolev class W p;1 loc . Therefore, Eq. (2.1.2) can be written in the classical divergence form @xi .aij @xj %/ C @xj aij % bi % D 0; (with bi changed appropriately). Similar results hold in the parabolic case. The reader is warned that the list of references is far from complete. A vast literature on the subject is given in the recent monograph [11] presenting results of many authors. 2.1.2 Uniqueness of solutions In the elliptic case, there are examples of infinitely differentiable b on Rd with d 1 such that, for A D Id (the unit matrix), there are several different (hence infinitely many) probability solutions to the equation L D 0. Similarly, in the parabolic case, we have constructed examples of infinitely differ- entiable b.x/ on Rd with d 3 such that for A D Id the Cauchy problem with some initial probability measure has infinitely many probability solutions. Therefore, certain additional conditions rather than smoothness are needed. We have found con- ditions of this sort expressed in terms of integrability of the coefficients with respect to solutions and (an alternative set of assumptions) in terms of so-called Lyapunov functions. We have also found sufficient conditions for the uniqueness of integrable solutions. In particular, we have shown that the uniqueness conditions for the class I of integrable solutions essentially differ from those for the class P. For example, the following results have been obtained (see [11]). Theorem 2.1.1. The elliptic equation L D 0 has at most one probability solution in either of the following cases, assuming in these cases (for simplicity) that A is locally Lipschitz and b is locally bounded. (i) There is a non-negative function V 2 C2 .Rd / such that lim jxj!1 V.x/ D C1 and LV.x/ 6 CV.x/ for some C 0. (ii) aij 1 C jxj2 ; bi 1 C jxj 2 L1 ./ for some probability solution .
- 52. Kolmogorov operators and SPDEs 33 Note that (i) does not assume (and does not ensure) the existence of a probability solution, while (ii) being satisfied by some solution, guarantees that this solution is the only one. However, we do not know whether it can happen that the elliptic equation L D 0 with A D Id and smooth b has no probability solutions, but has a nonzero signed solution in the class of bounded measures (this is impossible when d D 1). It follows from the previous theorem that, in case of the unit diffusion matrix (or a nondegenerate Lipschitz diffusion matrix) and bounded b, a probability solution is unique (if it exists). However, the case of irregular A has not been studied. Even the case of bounded continuous nondegenerate A (and bounded b) has not been investi- gated. The problem of existence of solutions has been better studied. Here are parabolic analogues. Theorem 2.1.2. Let A.x/ be locally Lipschitz and nondegenerate and let b.x; t/ be locally bounded. Suppose that there is a positive function V 2 C2 .Rd / such that lim jxj!1 V.x/ D C1 and LV.x; t/ 6 C C CV.x/ for some C 0. Then, for any probability measure on Rd , there is at most one probability solution to the Cauchy problem, with initial condition . Theorem 2.1.3. Let A D Id, and let b be a locally bounded vector field. Then, for the uniqueness of a probability solution to the Cauchy problem, it suffices to have a function V 2 C2 .Rd / with lim jxj!1 V.x/ D C1 and jrV.x/j 6 C1 such that LV.x; t/ 6 C2; while for the uniqueness of an integrable solution the inequality LV.x; t/ C2 is sufficient. In the case of a radial function V , such conditions actually mean that, for the uniqueness of a probability solution, the quantity .b.x; t/; x/ should not tend to C1 too quickly, and for the uniqueness of an integrable solution, .b.x; t/; x/ should not tend to 1 too quickly. Such a function V is called a Lyapunov function. A challenging open problem in this area concerns the cases d D 1 and d D 2: it is still unknown whether in these cases, for A D Id and smooth b, the Cauchy problem with a probability initial data has at most one probability solution (as noted above, there are counterexamples for all d 3). We now proceed to less regular diffusion matrices. Let U.x; r/ denote the ball of radius r centred at x.
- 53. 34 M. Röckner Let g be a bounded function on RdC1 . We set O.g; R/ D sup .x;t/2RdC1 sup r6R r 2 jU.x; r/j 2 tCr2 Z t “ y;z2U.x;r/ jg.y; s/ g.z; s/j dy dz ds: If lim R!0 O.g; R/ D 0, the function g is said to belong to the class VMOx.RdC1 /. If g 2 VMOx.RdC1 /, then one can always assume that O.g; R/ 6 w.R/ for all R 0, where w is a continuous function on Œ0; C1/ and w.0/ D 0. The following condition on A is used in our uniqueness result in case of A of low regularity. (H1) for every ball U Rd , there exist numbers D .U / 0 and M D M.U / 0 such that .A.x; t/y; y/ jyj2 ; kA.x; t/k 6 M for all .x; t/ 2 U Œ0; T and y 2 Rd . Theorem 2.1.4. Suppose that aij 2 VMOx;loc.Rd Œ0; T / and that the matrix A D .aij / satisfies condition (H1). Then, the set M D f 2 PW aij ; bi 2 L1 .; Rd Œ0; T /g consists of at most one element. We emphasise that this theorem gives uniqueness not in the whole class of proba- bility solutions but only in its subclass specified by the integrability of the drift b (for uniformly bounded b, this condition holds automatically). Let us say a few words about the infinite-dimensional case which attracts many researchers due to applications in stochastic partial differential equations, infinite- dimensional diffusions, and mathematical physics. For simplicity, we consider equa- tions on the space of real sequences R1 (the countable power of the real line). This space is a complete separable metric space with the distance d.x; y/ D 1 X nD1 2 n min.1; jxn ynj/: Due to the special structure of this space, the main concepts (but not the results) connected with Fokker–Planck–Kolmogorov equations on it are very similar to the case of Rd . Namely, given Borel functions aij and bi on R1 such that the finite- dimensional submatrices .aij /i;j 6n are symmetric non-negative-definite, we say that a Borel probability measure on R1 satisfies the equation L D 0
- 54. Kolmogorov operators and SPDEs 35 with the operator L D 1 X i;j D1 aij @xi @xj C 1 X iD1 bi @xi if the functions aij , bi are -integrable and for every function ' in finitely many variables belonging to the corresponding class C1 b one has Z R1 L' d D 0: Obviously, the function L' is -integrable since the series becomes a finite sum of -integrable functions. Set b WD .bi /1 iD1. Similarly, one defines a parabolic equation and the corresponding Cauchy prob- lem. As in the finite-dimensional case, interesting problems arise already for aij D ıij (which in Rd would mean the unit diffusion matrix). For example, if b.x/ D x, the standard Gaussian measure on R1 (the countable power of the standard Gaussian measure on the real line) is a solution to the corresponding equation. This measure is the only probability solution. Indeed, for a general b, the projection of any solution to Rn , denoted by n, satisfies the equation on Rn whose diffusion coefficients aij n and drift coefficients bi n are the conditional expectations of the functions aij and bi with i; j 6 n with respect to the measure and the -algebra Bn generated by the first n coordinates. This is obvious from the definitions. Therefore, in case of constant aij , we have aij n D aij and, in case of bi with i 6 n depending on x1; : : : ; xn, we have bi n D bi . In particular, for bi .x/ D xi , we have bi n.x/ D xi whenever i 6 n. Therefore, in the situation under consideration, the projection of any probability solution to Rn satisfies the same equation as the standard Gaussian measure on Rn , but this equation admits only one probability solution. However, the situation may change for other b. Actually, examples of linear b are known such that the corresponding equation (with aij D ıij ) has several different probability solutions that are Gaussian measures (hence there exist also non-Gaussian solutions, their convex combinations). A relatively simple case arises if we take b.x/ D x C v.x/; where the perturbation v takes values in the Hilbert space H D l2 and is bounded in the usual l2 -norm. In this case, it is known that every probability solution to the elliptic equation L D 0 is absolutely continuous with respect to and is unique (it is also known that a solution exists under the stated assumptions). Certainly, the assumption that v is a bounded l2 -valued perturbation is very restrictive. As a typical result on uniqueness in infinite dimensions, we mention the following theorem from [9]. Let us consider the following Cauchy problem. Let B D .Bi / be a sequence of Borel functions on R1 .0; T0/, where T0 0 is fixed, and let aij be Borel functions on R1 .0; T0/. Let us consider the Cauchy
- 55. Other documents randomly have different content
- 56. [41] English As We Speak It In Ireland, p. 77. [42] The Science of Language, vol. i, p. 166. [43] The last stand of the distinct -ed was made in Addison's day. He was in favor of retaining it, and in the Spectator for Aug. 4, 1711, he protested against obliterating the syllable in the termination of our praeter perfect tense, as in these words, drown'd, walk'd, arriv'd, for drowned, walked, arrived, which has very much disfigured the tongue, and turned a tenth part of our smoothest words into so many clusters of consonants. [44] A New English Grammar, pt. i, p. 380. [45] History of the English Language, p. 398. [46] And still more often as an adjective, as in it was a boughten dress. [47] You Know Me Al, p. 180; see also p. 122. [48] Cf. Lounsbury: History of the English Language, pp. 393 et seq. [49] Remark of a policeman talking to another. What he actually said was before the Elks was c'm 'ere. Come and here were one word, approximately cmear. The context showed that he meant to use the past perfect tense. [50] These examples are from Lardner's story, A New Busher Breaks In, in You Know Me Al, pp. 122 et seq. [51] You Know Me Al, op. cit., p. 124. [52] The Making of English, p. 53. [53] Cf. Dialect Notes, vol. iii, pt. i, p. 59; ibid., vol. III, pt. iv, p. 283. [54] Henry Bradley, in The Making of English, pp. 54-5: In the parts of England which were largely inhabited by Danes the native pronouns (i. e., heo, his, heom and heora) were supplanted by the Scandinavian pronouns which are represented by the modern she, they, them and their. This substitution, at first dialectical, gradually spread to the whole language. [55] Cf. Sweet: A New English Grammar, pt. i, p. 344, par. 1096. [56] Before a noun beginning with a vowel thine and mine are commonly substituted for thy and my, as in thine eyes and mine infirmity. But this is solely for the sake of euphony. There is no compensatory use of my and thy in the absolute. [57] The Making of English, p. 58. [58] Cf. The Dialect of Southeastern Missouri, by D. S. Crumb, Dialect Notes, vol. ii, pt. iv, 1903, p. 337. [59] It occurs, too, of course, in other dialects of English, though by no means in all. The Irish influence probably had something to do with its
- 57. prosperity in vulgar American. At all events, the Irish use it in the American manner. Joyce, in English As We Speak It in Ireland, pp. 34-5, argues that this usage was suggested by Gaelic. In Gaelic the accusative pronouns, e, i and iad (=him, her and them) are often used in place of the nominatives, sé, si and siad (=he, she and they), as in is iad sin na buachaillidhe (=them are the boys). This is good grammar in Gaelic, and the Irish, when they began to learn English, translated the locution literally. The familiar Irish John is dead and him always so hearty shows the same influence. [60] Pp. 144-50. [61] Modern English, p. 300. [62] A New English Grammar, pt. i, p. 339. [63] History of the English Language, pp. 274-5. [64] Modern English, p. 288-9. [65] Cf. p. 145n. [66] A New English Grammar, pt. i, p. 341. [67] It may be worth noting here that the misuse of me for my, as in I lit me pipe is quite unknown in American, either standard or vulgar. Even me own is seldom heard. This boggling of the cases is very common in spoken English. [68] A New English Grammar, pt. i, p. 341. [69] The King's English, p. 63. [70] Hon. Edward E. Browne, of Wisconsin, in the House of Representatives, July 18, 1918, p. 9965. [71] Cf. Vogue Affixes in Present-Day Word-Coinage, by Louise Pound, Dialect Notes, vol. v, pt. i, 1918. [72] The Speech of a Child Two Years of Age, Dialect Notes, vol. iv, pt. ii, 1914. [73] A New English Grammar, pt. i, pp. 437-8. [74] The King's English, p. 322. See especially the quotation from Frederick Greenwood, the distinguished English journalist. [75] Report of Edward J. Brundage, attorney-general of Illinois, on the East St. Louis massacre, Congressional Record, Jan. 7, 1918, p. 661. [76] The King's English, op. cit. [77] Oct. 1, 1864. [78] At all, by the way, is often displaced by any or none, as in he don't lover her any and it didn't hurt me none. [79] See the bibliography for the publication of Drs. Read and Pound.
- 58. [80] The only book that I can find definitely devoted to American sounds is A Handbook of American Speech, by Calvin L. Lewis; Chicago, 1916. It has many demerits. For example, the author gives a z-sound to the s in venison (p. 52). This is surely not American. [81] Maryland edition, July 18, 1914, p. 1. [82] Cf. Lounsbury: The Standard of Pronunciation in English, p. 172 et seq. [83] Stomp is used only in the sense of to stamp with the foot. One always stamps a letter. An analogue of stomp, accepted in correct English, is strop (e. g., razor-strop), from strap. [84] Our Own, Our Native Speech, McClure's Magazine, Oct., 1916. [Pg242] toc
- 59. VII Differences in Spelling § 1 Typical Forms —Some of the salient differences between American and English spelling are shown in the following list of common words: American English Anemia anaemia aneurism aneurysm annex (noun) annexe arbor arbour armor armour asphalt asphalte ataxia ataxy ax axe balk (verb) baulk baritone barytone bark (ship) barque behavior behaviour behoove behove buncombe bunkum burden (ship's) burthen cachexia cachexy
- 60. caliber calibre candor candour center centre check (bank) cheque checkered chequered cider cyder clamor clamour clangor clangour cloture closure[1] color colour connection connexion councilor councillor counselor counsellor cozy cosy curb kerb cyclopedia cyclopaedia defense defence demeanor demeanour diarrhea diarrhoea draft (ship's) draught dreadnaught dreadnought dryly drily ecology oecology ecumenical oecumenical edema oedema encyclopedia encyclopaedia endeavor endeavour eon aeon epaulet epaulette esophagus oesophagus
- 61. fagot faggot favor favour favorite favourite fervor fervour flavor flavour font (printer's) fount foregather forgather forego forgo form (printer's) forme fuse fuze gantlet (to run the—) gauntlet glamor glamour good-by good-bye gram gramme gray grey harbor harbour honor honour hostler ostler humor humour inclose enclose indorse endorse inflection inflexion inquiry enquiry jail gaol jewelry jewellery jimmy (burglar's) jemmy labor labour laborer labourer liter litre maneuver manoeuvre
- 62. medieval mediaeval meter metre misdemeanor misdemeanour mold mould mollusk mollusc molt moult mustache moustache neighbor neighbour neighborhood neighbourhood net (adj.) nett odor odour offense offence pajamas pyjamas parlor parlour peas (plu. of pea) pease picket (military) piquet plow plough pretense pretence program programme pudgy podgy pygmy pigmy rancor rancour rigor rigour rumor rumour savory savoury scimitar scimetar septicemia septicaemia show (verb) shew siphon syphon siren syren
- 63. skeptic sceptic slug (verb) slog slush slosh splendor splendour stanch staunch story (of a house) storey succor succour taffy toffy tire (noun) tyre toilet toilette traveler traveller tumor tumour valor valour vapor vapour veranda verandah vial phial vigor vigour vise (a tool) vice wagon waggon woolen woollen § 2 General Tendencies —This list is by no means exhaustive. According to a recent writer upon the subject, there are 812 words in which the prevailing American spelling differs from the English.[2] But enough examples are given to reveal a number of definite tendencies. American, in general, moves toward simplified forms of spelling more rapidly than English, and has got much further along the road. Redundant and
- 64. unnecessary letters have been dropped from whole groups of words —the u from the group of nouns in -our, with the sole exception of Saviour, and from such words as mould and baulk; the e from annexe, asphalte, axe, forme, pease, storey, etc.; the duplicate consonant from waggon, nett, faggot, woollen, jeweller, councillor, etc., and the silent foreign suffixes from toilette, epaulette, programme, verandah, etc. In addition, simple vowels have been substituted for degenerated diphthongs in such words as anaemia, [Pg246] oesophagus, diarrhoea and mediaeval, most of them from the Greek. Further attempts in the same direction are to be seen in the substitution of simple consonants for compound consonants, as in plow, bark, check, vial and draft; in the substitution of i for y to bring words into harmony with analogues, as in tire, cider and baritone (cf. wire, rider, merriment), and in the general tendency to get rid of the somewhat uneuphonious y, as in ataxia and pajamas. Clarity and simplicity are also served by substituting ct for x in such words as connection and inflection, and s for c in words of the defense group. The superiority of jail to gaol is made manifest by the common mispronunciation of the latter, making it rhyme with coal. The substitution of i for e in such words as indorse, inclose and jimmy is of less patent utility, but even here there is probably a slight gain in euphony. Of more obscure origin is what seems to be a tendency to avoid the o-sound, so that the English slog becomes slug, podgy becomes pudgy, nought becomes naught, slosh becomes slush, toffy becomes taffy, and so on. Other changes carry their own justification. Hostler is obviously better American than ostler, though it may be worse English. Show is more logical than shew.[3] Cozy is more nearly phonetic than cosy. Curb has analogues in curtain, curdle, curfew, curl, currant, curry, curve, curtsey, curse, currency, cursory, curtail, cur, curt and many other common words: kerb has very few, and of them only kerchief and kernel are in general use. Moreover, the English themselves use curb as a verb and in all noun senses save that shown in kerbstone.
- 65. But a number of anomalies remain. The American substitution of a for e in gray is not easily explained, nor is the substitution of k for c in skeptic and mollusk, nor the retention of e in forego, nor the unphonetic substitution of s for z in fuse, [Pg247] nor the persistence of the first y in pygmy. Here we have plain vagaries, surviving in spite of attack by orthographers. Webster, in one of his earlier books, denounced the k in skeptic as a mere pedantry, but later on he adopted it. In the same way pygmy, gray and mollusk have been attacked, but they still remain sound American. The English themselves have many more such illogical forms to account for. In the midst of the our-words they cling to a small number in or, among them, stupor. Moreover, they drop the u in many derivatives, for example, in arboreal, armory, clamorously, clangorous, odoriferous, humorist, laborious and rigorism. If it were dropped in all derivatives the rule would be easy to remember, but it is retained in some of them, for example, colourable, favourite, misdemeanour, coloured and labourer. The derivatives of honour exhibit the confusion clearly. Honorary, honorarium and honorific drop the u, but honourable retains it. Furthermore, the English make a distinction between two senses of rigor. When used in its pathological sense (not only in the Latin form of rigor mortis, but as an English word) it drops the u; in all other senses it retains the u. The one American anomaly in this field is Saviour. In its theological sense it retains the u; but in that sense only. A sailor who saves his ship is its savior, not its saviour. § 3 The Influence of Webster —At the time of the first settlement of America the rules of English orthography were beautifully vague, and so we find the early documents full of spellings that would give an English lexicographer much pain today. Now and then a curious foreshadowing of later American usage is encountered. On July 4, 1631, for example, John
- 66. Winthrop wrote in his journal that the governour built a bark at Mistick, which was launched this day. But during the eighteenth century, and especially after the publication of Johnson's dictionary, there was a general movement in England toward a more inflexible orthography, and many hard and fast rules, still surviving, were then laid down. It was Johnson himself who [Pg248] established the position of the u in the our words. Bailey, Dyche and the other lexicographers before him were divided and uncertain; Johnson declared for the u, and though his reasons were very shaky[4] and he often neglected his own precept, his authority was sufficient to set up a usage which still defies attack in England. Even in America this usage was not often brought into question until the last quarter of the eighteenth century. True enough, honor appears in the Declaration of Independence, but it seems to have got there rather by accident than by design. In Jefferson's original draft it is spelled honour. So early as 1768 Benjamin Franklin had published his Scheme for a New Alphabet and a Reformed Mode of Spelling, with Remarks and Examples Concerning the Same, and an Enquiry Into its Uses and induced a Philadelphia typefounder to cut type for it, but this scheme was too extravagant to be adopted anywhere, or to have any appreciable influence upon spelling.[5] It was Noah Webster who finally achieved the divorce between English example and American practise. He struck the first blow in his Grammatical Institute of the English Language, published at Hartford in 1783. Attached to this work was an appendix bearing the formidable title of An Essay on the Necessity, Advantages and Practicability of Reforming the Mode of Spelling, and of Rendering the Orthography of Words Correspondent to the Pronunciation, and during the same year, at Boston, he set forth his ideas a second time in the first edition of his American Spelling Book. The influence of this spelling book was immediate and profound. It took the place in the schools of Dilworth's Aby-sel-pha, the favorite of the generation preceding, and maintained its authority for fully a century. Until Lyman Cobb entered the lists with his New Spelling Book, in 1842, its innumerable editions scarcely had [Pg249] any
- 67. rivalry, and even then it held its own. I have a New York edition, dated 1848, which contains an advertisement stating that the annual sale at that time was more than a million copies, and that more than 30,000,000 copies had been sold since 1783. In the late 40's the publishers, George F. Cooledge Bro., devoted the whole capacity of the fastest steam press in the United States to the printing of it. This press turned out 525 copies an hour, or 5,250 a day. It was constructed expressly for printing Webster's Elementary Spelling Book [the name had been changed in 1829] at an expense of $5,000. Down to 1889, 62,000,000 copies of the book had been sold. The appearance of Webster's first dictionary, in 1806, greatly strengthened his influence. The best dictionary available to Americans before this was Johnson's in its various incarnations, but against Johnson's stood a good deal of animosity to its compiler, whose implacable hatred of all things American was well known to the citizens of the new republic. John Walker's dictionary, issued in London in 1791, was also in use, but not extensively. A home-made school dictionary, issued at New Haven in 1798 or 1799 by one Samuel Johnson, Jr.—apparently no relative of the great Sam—and a larger work published a year later by Johnson and the Rev. John Elliott, pastor in East Guilford, Conn., seem to have made no impression, despite the fact that the latter was commended by Simeon Baldwin, Chauncey Goodrich and other magnificoes of the time and place, and even by Webster himself. The field was thus open to the laborious and truculent Noah. He was already the acknowledged magister of lexicography in America, and there was an active public demand for a dictionary that should be wholly American. The appearance of his first duodecimo, according to Williams,[6] thereby took on something of the character of a national event. It was received, not critically, but patriotically, and its imperfections were swallowed as eagerly as its merits. Later on Webster had to meet formidable critics, at home as well as abroad, but for nearly a quarter of a century he reigned almost unchallenged. Edition after edition of his dictionary was published,
- 68. [Pg250] each new one showing additions and improvements. Finally, in 1828, he printed his great American Dictionary of the English Language, in two large octavo volumes. It held the field for half a century, not only against Worcester and the other American lexicographers who followed him, but also against the best dictionaries produced in England. Until very lately, indeed, America remained ahead of England in practical dictionary making. Webster had declared boldly for simpler spellings in his early spelling books; in his dictionary of 1806 he made an assault at all arms upon some of the dearest prejudices of English lexicographers. Grounding his wholesale reforms upon a saying by Franklin, that those people spell best who do not know how to spell—i. e., who spell phonetically and logically—he made an almost complete sweep of whole classes of silent letters—the u in the -our words, the final e in determine and requisite, the silent a in thread, feather and steady, the silent b in thumb, the s in island, the o in leopard, and the redundant consonants in traveler, wagon, jeweler, etc. (English: traveller, waggon, jeweller). More, he lopped the final k from frolick, physick and their analogues. Yet more, he transposed the e and the r in all words ending in re, such as theatre, lustre, centre and calibre. Yet more, he changed the c in all words of the defence class to s. Yet more, he changed ph to f in words of the phantom class, ou to oo in words of the group class, ow to ou in crowd, porpoise to porpess, acre to aker, sew to soe, woe to wo, soot to sut, gaol to jail, and plough to plow. Finally, he antedated the simplified spellers by inventing a long list of boldly phonetic spellings, ranging from tung for tongue to wimmen for women, and from hainous for heinous to cag for keg. A good many of these new spellings, of course, were not actually Webster's inventions. For example, the change from -our to -or in words of the honor class was a mere echo of an earlier English usage, or, more accurately, of an earlier English uncertainty. In the first three folios of Shakespeare, 1623, 1632 and 1663-6, honor and honour were used indiscriminately and in almost equal proportions; English spelling was still fluid, and [Pg251] the -our-form was not
- 69. consistently adopted until the fourth folio of 1685. Moreover, John Wesley, the founder of Methodism, is authority for the statement that the -or-form was a fashionable impropriety in England in 1791. But the great authority of Johnson stood against it, and Webster was surely not one to imitate fashionable improprieties. He deleted the u for purely etymological reasons, going back to the Latin honor, favor and odor without taking account of the intermediate French honneur, faveur and odeur. And where no etymological reasons presented themselves, he made his changes by analogy and for the sake of uniformity, or for euphony or simplicity, or because it pleased him, one guesses, to stir up the academic animals. Webster, in fact, delighted in controversy, and was anything but free from the national yearning to make a sensation. A great many of his innovations, of course, failed to take root, and in the course of time he abandoned some of them himself. In his early Essay on the Necessity, Advantage and Practicability of Reforming the Mode of Spelling he advocated reforms which were already discarded by the time he published the first edition of his dictionary. Among them were the dropping of the silent letter in such words as head, give, built and realm, making them hed, giv, bilt and relm; the substitution of doubled vowels for decayed diphthongs in such words as mean, zeal and near, making them meen, zeel and neer; and the substitution of sh for ch in such French loan-words as machine and chevalier, making them masheen and shevaleer. He also declared for stile in place of style, and for many other such changes, and then quietly abandoned them. The successive editions of his dictionary show still further concessions. Croud, fether, groop, gillotin, iland, insted, leperd, soe, sut, steddy, thret, thred, thum and wimmen appear only in the 1806 edition. In 1828 he went back to crowd, feather, group, island, instead, leopard, sew, soot, steady, thread, threat, thumb and women, and changed gillotin to guillotin. In addition, he restored the final e in determine, discipline, requisite, imagine, etc. In 1838, revising his dictionary, he abandoned a good many spellings that had appeared in either the 1806 or the 1828 edition, notably maiz for maize, [Pg252] suveran for sovereign and
- 70. guillotin for guillotine. But he stuck manfully to a number that were quite as revolutionary—for example, aker for acre, cag for keg, grotesk for grotesque, hainous for heinous, porpess for porpoise and tung for tongue—and they did not begin to disappear until the edition of 1854, issued by other hands and eleven years after his death. Three of his favorites, chimist for chemist, neger for negro and zeber for zebra, are incidentally interesting as showing changes in American pronunciation. He abandoned zeber in 1828, but remained faithful to chimist and neger to the last. But though he was thus forced to give occasional ground, and in more than one case held out in vain, Webster lived to see the majority of his reforms adopted by his countrymen. He left the ending in -or triumphant over the ending in -our, he shook the security of the ending in -re, he rid American spelling of a great many doubled consonants, he established the s in words of the defense group, and he gave currency to many characteristic American spellings, notably jail, wagon, plow, mold and ax. These spellings still survive, and are practically universal in the United States today; their use constitutes one of the most obvious differences between written English and written American. Moreover, they have founded a general tendency, the effects of which reach far beyond the field actually traversed by Webster himself. New words, and particularly loan-words, are simplified, and hence naturalized in American much more quickly than in English. Employé has long since become employee in our newspapers, and asphalte has lost its final e, and manoeuvre has become maneuver, and pyjamas has become pajamas. Even the terminology of science is simplified and Americanized. In medicine, for example, the highest American usage countenances many forms which would seem barbarisms to an English medical man if he encountered them in the Lancet. In derivatives of the Greek haima it is the almost invariable American custom to spell the root syllable hem, but the more conservative English make it haem—e. g., in haemorrhage and haemiplegia. In an exhaustive list of diseases issued by the United States Public Health [Pg253] Service[7] the haem-form does not appear once. In the same
- 71. way American usage prefers esophagus, diarrhea and gonorrhea to the English oesophagus, diarrhoea and gonorrhoea. In the style- book of the Journal of the American Medical Association[8] I find many other spellings that would shock an English medical author, among them curet for curette, cocain for cocaine, gage for gauge, intern for interne, lacrimal for lachrymal, and a whole group of words ending in -er instead of in -re. Webster's reforms, it goes without saying, have not passed unchallenged by the guardians of tradition. A glance at the literature of the first years of the nineteenth century shows that most of the serious authors of the time ignored his new spellings, though they were quickly adopted by the newspapers. Bancroft's Life of Washington contains -our endings in all such words as honor, ardor and favor. Washington Irving also threw his influence against the -or ending, and so did Bryant and most of the other literary big-wigs of that day. After the appearance of An American Dictionary of the English Language, in 1828, a formal battle was joined, with Lyman Cobb and Joseph E. Worcester as the chief opponents of the reformer. Cobb and Worcester, in the end, accepted the -or ending and so surrendered on the main issue, but various other champions arose to carry on the war. Edward S. Gould, in a once famous essay, [9] denounced the whole Websterian orthography with the utmost fury, and Bryant, reprinting this philippic in the Evening Post, said that on account of Webster the English language has been undergoing a process of corruption for the last quarter of a century, and offered to contribute to a fund to have Gould's denunciation read twice a year in every school-house in the United States, until every trace of Websterian spelling disappears from the land. But Bryant was forced to admit that, even in 1856, the chief novelties of the Connecticut school-master who taught millions to read but not one to sin were [Pg254] adopted and propagated by the largest publishing house, through the columns of the most widely circulated monthly magazine, and through one of the ablest and most widely circulated newspapers in the United States—which is to say, the Tribune under Greeley. The last academic attack was delivered by
- 72. Bishop Coxe in 1886, and he contented himself with the resigned statement that Webster has corrupted our spelling sadly. Lounsbury, with his active interest in spelling reform, ranged himself on the side of Webster, and effectively disposed of the controversy by showing that the great majority of his spellings were supported by precedents quite as respectable as those behind the fashionable English spellings. In Lounsbury's opinion, a good deal of the opposition to them was no more than a symptom of antipathy to all things American among certain Englishmen and of subservience to all things English among certain Americans.[10] Webster's inconsistency gave his opponents a formidable weapon for use against him—until it began to be noticed that the orthodox English spelling was quite as inconsistent. He sought to change acre to aker, but left lucre unchanged. He removed the final f from bailiff, mastiff, plaintiff and pontiff, but left it in distaff. He changed c to s in words of the offense class, but left the c in fence. He changed the ck in frolick, physick, etc., into a simple c, but restored it in such derivatives as frolicksome. He deleted the silent u in mould, but left it in court. These slips were made the most of by Cobb in a pamphlet printed in 1831.[11] He also detected Webster in the frequent faux pas of using spellings in his definitions and explanations that conflicted with the spellings he advocated. Various other purists joined in the attack, and it was renewed with great fury after the appearance of Worcester's dictionary, in 1846. Worcester, who had begun his lexicographical labors by editing Johnson's dictionary, was a good deal more conservative than Webster, and so the partisans of conformity rallied around him, and for [Pg255] a while the controversy took on all the rancor of a personal quarrel. Even the editions of Webster printed after his death, though they gave way on many points, were violently arraigned. Gould, in 1867, belabored the editions of 1854 and 1866,[12] and complained that for the past twenty-five years the Websterian replies have uniformly been bitter in tone, and very free in the imputation of personal motives, or interested or improper motives, on the part of opposing critics. At this time Webster himself had been dead for twenty-two
- 73. years. Schele de Vere, during the same year, denounced the publishers of the Webster dictionaries for applying immense capital and a large stock of energy and perseverance to the propagation of his new and arbitrarily imposed orthography.[13] § 4 Exchanges —As in vocabulary and in idiom, there are constant exchanges between English and American in the department of orthography. Here the influence of English usage is almost uniformly toward conservatism, and that of American usage is as steadily in the other direction. The logical superiority of American spelling is well exhibited by its persistent advance in the face of the utmost hostility. The English objection to our simplifications, as Brander Matthews points out, is not wholly or even chiefly etymological; its roots lie, to borrow James Russell Lowell's phrase, in an esthetic hatred burning with as fierce a flame as ever did theological hatred. There is something inordinately offensive to English purists in the very thought of taking lessons from this side of the water, particularly in the mother tongue. The opposition, transcending the academic, takes on the character of the patriotic. Any American, continues Matthews, who chances to note the force and the fervor and the frequency of the objurgations against American spelling in the columns of the Saturday Review, for example, and of the Athenaeum, may find himself wondering as to the date of the [Pg256] papal bull which declared the infallibility of contemporary British orthography, and as to the place where the council of the Church was held at which it was made an article of faith.[14] This was written more than a quarter of a century ago. Since then there has been a lessening of violence, but the opposition still continues. No self-respecting English author would yield up the -our ending for an instant, or write check for cheque, or transpose the last letters in the -re words.
- 74. Nevertheless, American spelling makes constant gains across the water, and they more than offset the occasional fashions for English spellings on this side. Schele de Vere, in 1867, consoled himself for Webster's arbitrarily imposed orthography by predicting that it could be only temporary—that, in the long run, North America depends exclusively on the mother-country for its models of literature. But the event has blasted this prophecy and confidence, for the English, despite their furious reluctance, have succumbed to Webster more than once. The New English Dictionary, a monumental work, shows many silent concessions, and quite as many open yieldings—for example, in the case of ax, which is admitted to be better than axe on every ground. Moreover, English usage tends to march ahead of it, outstripping the liberalism of its editor, Sir James A. H. Murray. In 1914, for example, Sir James was still protesting against dropping the first e from judgement, a characteristic Americanism, but during the same year the Fowlers, in their Concise Oxford Dictionary, put judgment ahead of judgement; and two years earlier the Authors' and Printers' Dictionary, edited by Horace Hart, [15] had dropped judgement altogether. Hart is Controller of the Oxford University Press, and the Authors' and Printers' Dictionary is an authority accepted by nearly all of the great English book publishers and newspapers. Its last edition shows a great many American spellings. For example, it recommends the use of jail and jailer in place [Pg257] of the English gaol and gaoler, says that ax is better than axe, drops the final e from asphalte and forme, changes the y to i in cyder, cypher and syren and advocates the same change in tyre, drops the redundant t from nett, changes burthen to burden, spells wagon with one g, prefers fuse to fuze, and takes the e out of storey. Rules for Compositors and Readers at the University Press, Oxford, also edited by Hart (with the advice of Sir James Murray and Dr. Henry Bradley), is another very influential English authority. [16] It gives its imprimatur to bark (a ship), cipher, siren, jail, story, tire and wagon, and even advocates kilogram and omelet. Finally, there is Cassell's English Dictionary.[17] It clings to the -our and -re endings and to annexe, waggon and cheque, but it prefers jail to
- 75. gaol, net to nett, asphalt to asphalte and story to storey, and comes out flatly for judgment, fuse and siren. Current English spelling, like our own, shows a number of uncertainties and inconsistencies, and some of them are undoubtedly the result of American influences that have not yet become fully effective. The lack of harmony in the -our words, leading to such discrepancies as honorary and honourable, I have already mentioned. The British Board of Trade, in attempting to fix the spelling of various scientific terms, has often come to grief. Thus it detaches the final -me from gramme in such compounds as kilogram and milligram, but insists upon gramme when the word stands alone. In American usage gram is now common, and scarcely challenged. All the English authorities that I have consulted prefer metre and calibre to the American meter and caliber.[18] They also support the ae in such words as aetiology, aesthetics, mediaeval and anaemia, and the oe in oesophagus, [Pg258] manoeuvre and diarrhoea. They also cling to such forms as mollusc, kerb, pyjamas and ostler, and to the use of x instead of ct in connexion and inflexion. The Authors' and Printers' Dictionary admits the American curb, but says that the English kerb is more common. It gives barque, plough and fount, but grants that bark, plow and font are good in America. As between inquiry and enquiry, it prefers the American inquiry to the English enquiry, but it rejects the American inclose and indorse in favor of the English enclose and endorse.[19] Here American spelling has driven in a salient, but has yet to take the whole position. A number of spellings, nearly all American, are trembling on the brink of acceptance in both countries. Among them is rime (for rhyme). This spelling was correct in England until about 1530, but its recent revival was of American origin. It is accepted by the Oxford Dictionary and by the editors of the Cambridge History of English Literature, but it seldom appears in an English journal. The same may be said of grewsome. It has got a footing in both countries, but the weight of English opinion is still against it. Develop (instead of develope) has gone further in both countries. So has engulf, for engulph. So has gipsy for gypsy.
- 76. American imitation of English orthography has two impulses behind it. First, there is the colonial spirit, the desire to pass as English—in brief, mere affectation. Secondly, there is the wish among printers, chiefly of books and periodicals, to reach a compromise spelling acceptable in both countries, thus avoiding expensive revisions in case of republication in England.[20] [Pg259] The first influence need not detain us. It is chiefly visible among folk of fashionable pretensions, and is not widespread. At Bar Harbor, in Maine, some of the summer residents are at great pains to put harbour instead of harbor on their stationery, but the local postmaster still continues to stamp all mail Bar Harbor, the legal name of the place. In the same way American haberdashers sometimes advertise pyjamas instead of pajamas, just as they advertise braces instead of suspenders and vests instead of undershirts. But this benign folly does not go very far. Beyond occasionally clinging to the -re ending in words of the theatre group, all American newspapers and magazines employ the native orthography, and it would be quite as startling to encounter honour or jewellery in one of them as it would be to encounter gaol or waggon. Even the most fashionable jewelers in Fifth avenue still deal in jewelry, not in jewellery. The second influence is of more effect and importance. In the days before the copyright treaty between England and the United States, one of the standing arguments against it among the English was based upon the fear that it would flood England with books set up in America, and so work a corruption of English spelling.[21] This fear, as we have seen, had a certain plausibility; there is not the slightest doubt that American books and American magazines have done valiant missionary service for American orthography. But English conservatism still holds out stoutly enough to force American printers to certain compromises. When a book is designed for circulation in both countries it is common for the publisher to instruct the printer to employ English spelling. This English spelling, at the Riverside Press,[22] embraces all the -our endings and the following further forms:
- 77. cheque chequered connexion dreamt faggot forgather forgo grey inflexion jewellery leapt premises (in logic) waggon It will be noted that gaol, tyre, storey, kerb, asphalte, annexe, ostler, mollusc and pyjamas are not listed, nor are the words ending in -re. These and their like constitute the English contribution to the compromise. Two other great American book presses, that of the Macmillan Company[23] and that of the J. S. Cushing Company,[24] add gaol and storey to the list, and also behove, briar, drily, enquire, gaiety, gipsy, instal, judgement, lacquey, moustache, nought, pigmy, postillion, reflexion, shily, slily, staunch and verandah. Here they go too far, for, as we have seen, the English themselves have begun to abandon briar, enquire and judgement. Moreover, lacquey is going out over there, and gipsy is not English, but American. The Riverside Press, even in books intended only for America, prefers certain English forms, among them, anaemia, axe, mediaeval, mould, plough, programme and quartette, but in compensation it stands by such typical Americanisms as caliber, calk, center, cozy, defense, foregather, gray, hemorrhage, luster, maneuver, mustache, theater
- 78. and woolen. The Government Printing Office at Washington follows Webster's New International Dictionary,[25] which supports most of the innovations of Webster himself. This dictionary is the authority in perhaps a majority of American printing offices, with the Standard and the Century supporting it. The latter two also follow Webster, notably in his -er [Pg261] endings and in his substitution of s for c in words of the defense class. The Worcester Dictionary is the sole exponent of English spelling in general circulation in the United States. It remains faithful to most of the -re endings, and to manoeuvre, gramme, plough, sceptic, woollen, axe and many other English forms. But even Worcester favors such characteristic American spellings as behoove, brier, caliber, checkered, dryly, jail and wagon. § 5 Simplified Spelling —The current movement toward a general reform of English- American spelling is of American origin, and its chief supporters are Americans today. Its actual father was Webster, for it was the long controversy over his simplified spellings that brought the dons of the American Philological Association to a serious investigation of the subject. In 1875 they appointed a committee to inquire into the possibility of reform, and in 1876 this committee reported favorably. During the same year there was an International Convention for the Amendment of English Orthography at Philadelphia, with several delegates from England present, and out of it grew the Spelling Reform Association.[26] In 1878 a committee of American philologists began preparing a list of proposed new spellings, and two years later the Philological Society of England joined in the work. In 1883 a joint manifesto was issued, recommending various general simplifications. In 1886 the American Philological Association issued independently a list of recommendations affecting about 3,500 words, and falling under ten headings. Practically all of the changes
- 79. proposed had been put forward 80 years before by Webster, and some of them had entered into unquestioned American usage in the meantime, e. g., the deletion of the u from the -our words, the substitution of [Pg262] er for re at the end of words, the reduction of traveller to traveler, and the substitution of z for s wherever phonetically demanded, as in advertize and cozy. The trouble with the others was that they were either too uncouth to be adopted without a struggle or likely to cause errors in pronunciation. To the first class belonged tung for tongue, ruf for rough, batl for battle and abuv for above, and to the second such forms as cach for catch and troble for trouble. The result was that the whole reform received a set-back: the public dismissed the industrious professors as a pack of dreamers. Twelve years later the National Education Association revived the movement with a proposal that a beginning be made with a very short list of reformed spellings, and nominated the following by way of experiment: tho, altho, thru, thruout, thoro, thoroly, thorofare, program, prolog, catalog, pedagog and decalog. This scheme of gradual changes was sound in principle, and in a short time at least two of the recommended spellings, program and catalog, were in general use. Then, in 1906, came the organization of the Simplified Spelling Board, with an endowment of $15,000 a year from Andrew Carnegie, and a formidable membership of pundits. The board at once issued a list of 300 revised spellings, new and old, and in August, 1906, President Roosevelt ordered their adoption by the Government Printing Office. But this unwise effort to hasten matters, combined with the buffoonery characteristically thrown about the matter by Roosevelt, served only to raise up enemies, and since then, though it has prudently gone back to more discreet endeavors and now lays main stress upon the original 12 words of the National Education Association, the Board has not made a great deal of progress.[27] From time to time it issues impressive lists of newspapers and periodicals that are using some, at least, of its revised spellings and of colleges that have made them optional, but an inspection of these lists shows that very few [Pg263] publications of
- 80. any importance have been converted[28] and that most of the great universities still hesitate. It has, however, greatly reinforced the authority behind many of Webster's spellings, and it has done much to reform scientific orthography. Such forms as gram, cocain, chlorid, anemia and anilin are the products of its influence. Despite the large admixture of failure in this success there is good reason to believe that at least two of the spellings on the National Education Association list, tho and thru, are making not a little quiet progress. I read a great many manuscripts by American authors, and find in them an increasing use of both forms, with the occasional addition of altho, thoro and thoroly. The spirit of American spelling is on their side. They promise to come in as honor, bark, check, wagon and story came in many years ago, as tire,[29] esophagus and theater came in later on, as program, catalog and cyclopedia came in only yesterday, and as airplane (for aëroplane)[30] is coming in today. A constant tendency toward logic and simplicity is visible; if the spelling of English and American does not grow farther and farther apart it is only because American drags English along. There is incessant experimentalization. New forms appear, are tested, and then either gain general acceptance or disappear. One such, now struggling for recognition, is alright, a compound of all and right, made by analogy with already and almost. I find it in American manuscripts every day, and it not infrequently gets into print.[31] So far no dictionary supports it, but [Pg264] it has already migrated to England.[32] Meanwhile, one often encounters, in American advertising matter, such experimental forms as burlesk, foto, fonograph, kandy, kar, holsum, kumfort and Q-room, not to mention sulfur. Segar has been more or less in use for half a century, and at one time it threatened to displace cigar. At least one American professor of English predicts that such forms will eventually prevail. Even fosfate and fotograph, he says, are bound to be the spellings of the future.[33] § 6
- 81. Minor Differences —Various minor differences remain to be noticed. One is a divergence in orthography due to differences in pronunciation. Specialty, aluminum and alarm offer examples. In English they are speciality, aluminium and alarum, though alarm is also an alternative form. Specialty, in America, is always accented on the first syllable; speciality, in England, on the third. The result is two distinct words, though their meaning is identical. How aluminium, in America, lost its fourth syllable I have been unable to determine, but all American authorities now make it aluminum and all English authorities stick to aluminium. Another difference in usage is revealed in the spelling and pluralization of foreign words. Such words, when they appear in an English publication, even a newspaper, almost invariably bear the correct accents, but in the United States it is almost as invariably the rule to omit these accents, save in publications of considerable pretensions. This is notably the case with café crêpe, début, débutante, portière, levée, éclat, fête, régime, rôle, soirée, protégé, élite, mêlée, tête-à-tête and répertoire. It is rare to encounter any of them with its proper accents in an American newspaper; it is rare to encounter them unaccented in an English [Pg265] newspaper. This slaughter of the accents, it must be obvious, greatly aids the rapid naturalization of a newcomer. It loses much of its foreignness at once, and is thus easier to absorb. Dépôt would have been a long time working its way into American had it remained dépôt, but immediately it became plain depot it got in. The process is constantly going on. I often encounter naïveté without its accents, and even déshabille, hofbräu, señor and résumé. Cañon was changed to canyon years ago, and the cases of exposé, divorcée, schmierkäse, employé and matinée are familiar. At least one American dignitary of learning, Brander Matthews, has openly defended and even advocated this clipping of accents. In speaking of naïf and naïveté, which he welcomes because we have no exact equivalent for either word, he says: But they will need to shed their accents and to adapt themselves somehow to the traditions of
- 82. our orthography.[34] He goes on: After we have decided that the foreign word we find knocking at the doors of English [he really means American, as the context shows] is likely to be useful, we must fit it for naturalization by insisting that it shall shed its accents, if it has any; that it shall change its spelling, if this is necessary; that it shall modify its pronunciation, if this is not easy for us to compass; and that it shall conform to all our speech-habits, especially in the formation of the plural.[35] In this formation of the plural, as elsewhere, English regards the precedents and American makes new ones. All the English authorities that I have had access to advocate retaining the foreign plurals of most of the foreign words in daily use, e. g., sanatoria, appendices, virtuosi, formulae and libretti. But American usage favors plurals of native cut, and the Journal of the American Medical Association goes so far as to approve curriculums and septums. Banditti, in place of bandits, would seem an affectation in America, and so would soprani for sopranos [Pg266] and soli for solos.[36] The last two are common in England. Both English and American labor under the lack of native plurals for the two everyday titles, Mister and Missus. In the written speech, and in the more exact forms of the spoken speech, the French plurals, Messieurs and Mesdames, are used, but in the ordinary spoken speech, at least in America, they are avoided by circumlocution. When Messieurs has to be spoken it is almost invariably pronounced messers, and in the same way Mesdames becomes mez-dames, with the first syllable rhyming with sez and the second, which bears the accent, with games. In place of Mesdames a more natural form, Madames, seems to be gaining ground in America. Thus, I lately found Dames du Sacré Coeur translated as Madames of the Sacred Heart in a Catholic paper of wide circulation,[37] and the form is apparently used by American members of the community. In capitalization the English are a good deal more conservative than we are. They invariably capitalize such terms as Government, Prime Minister and Society, when used as proper nouns; they capitalize Press, Pulpit, Bar, etc., almost as often. In America a
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