Download - Free Documents

a
r
X
i
v
.
v
m
a
t
h
.
S
P
J
a
n
RANDOM COVARIANCE MATRICES UNIVERSALITY OF
LOCAL STATISTICS OF EIGENVALUES
TERENCE TAO AND VAN VU
Abstract. We study the eigenvalues values of the covariance matrix
n
M
M
of a large rectangular matrix M Mn,p
ij
ipjn
whose entries
are iid random variables of mean zero, variance one, and having nite C
th
moment for some suciently large constant C
.
The main result of this paper is a Four Moment Theorem for iid covariance
matrices analogous to the Four Moment Theorem for Wigner matrices estab
lished by the authors in see also . Indeed, our arguments here draw
heavily from those in . As in that paper, we can use this theorem together
with existing results to establish universality of local statistics of eigenvalues
under mild conditions.
As a byproduct of our arguments, we also extend our previous results on
random hermitian matrices to the case in which the entries have nite C
th
moment rather than exponential decay.
. Introduction
.. The model. The main purpose of this paper is to study the asymptotic local
eigenvalue statistics of covariance matrices of large random matrices. Let us rst
x the matrix ensembles that we will be studying.
Denition Random covariance matrices. Let n be a large integer parameter
going o to innity, and let p pn be another integer parameter such that p n
and lim
n
p/n y for some lt y . We let M M
n,p
ij
ip,jn
be a random p n matrix, whose distribution of course is allowed to depend on
n. We say that the matrix ensemble M obeys condition C with some exponent
C
if the random variables
ij
are jointly independent, have mean zero and
variance , and obey the moment condition sup
i,j
E
ij
C
C for some constant
C independent of n, p. We say that the matrix M is iid if the
ij
are identically
and independently distributed with law independent of n.
Given such a matrix, we form the nn covariance matrix W W
n,p
n
M
M.
This matrix has rank p and so the rst n p eigenvalues are trivial we order the
necessarily positive remaining eigenvalues of these matrices counting multiplic
ity as
W...
p
W.
T. Tao is supported by a grant from the MacArthur Foundation, by NSF grant DMS,
and by the NSF Waterman award.
V. Vu is supported by research grants DMS and AFOSARFA.
TERENCE TAO AND VAN VU
We often abbreviate
i
W as
i
.
Remark . In this paper we will focus primarily on the case y , but several
of our results extend to other values of y as well. The case p gt n can of course be
deduced from the p lt n case after some minor notational changes by transposing
the matrix M, which does not aect the nontrivial eigenvalues of the covariance
matrix. One can also easily normalise the variance of the entries to be some other
quantity
than if one wishes. Observe that the quantities
i
n
/
i
can
be interpreted as the nontrivial singular values of the original matrix M, and
,...,
p
can also be interpreted as the eigenvalues of the pp matrix
n
MM
. It
will be convenient to exploit all three of these spectral interpretations of
,...,
p
in this paper. Condition C is analogous to Condition C for Wignertype matrices
in , but with the exponential decay hypothesis relaxed to polynomial decay only.
The wellknown MarchenkoPastur law governs the bulk distribution of the eigen
values
,...,
p
of W
Theorem MarchenkoPastur law. Assume Condition C with C
, and
suppose that p/n y for some lt y . Then for any x gt , the random
variables
p
ip
i
Wx
converge in probability to
x
MP,y
x dx, where
MP,y
x
xy
b xx a
a,b
x
and
a
y
b
y
.
When furthermore M is iid, one can also obtain the case C
.
Proof. For C
, see , for C
gt , see for the C
iid case, see
. Further results are known on the rate of convergence see .
In this paper we are concerned instead with the local eigenvalue statistics. A model
case is the complex Wishart ensemble, in which the
ij
are iid variables which are
complex gaussians with mean zero and variance . In this case, the distribution of
the eigenvalues
,...,
n
of W can be explicitly computed as a special case of
the Laguerre unitary ensemble. For instance, when p n, the joint distribution is
given by the density function
n
,...,
n
cn
iltjn
i
j
exp
n
i
i
/
for some explicit normalization constant cn.
Very similarly to the GUE case, one can use this explicit formula to directly com
pute several local statistics, including the distribution of the largest and smallest
eigenvalues , the correlation functions etc. Also in similarity to the GUE
UNIVERSALITY FOR COVARIANCE MATRICES
case, it is widely conjectured that these statistics hold for a much larger class of
random matrices. For some earlier results in this direction, we refer to , , ,
and the references therein.
The goal of this paper is to establish a Four Moment theorem for random covari
ance matrices, as an analogue of a recent result in . This theorem asserts that
all local statistics of the eigenvalues of W
n
is determined by the rst four moments
of the entries.
.. The Four Moment Theorem. We rst need some denitions.
Denition Frequent events. Let E be an event depending on n.
E holds asymptotically almost surely if PE o.
E holds with high probability if PE On
c
for some constant c gt
independent of n.
E holds with overwhelming probability if PE O
C
n
C
for every
constant C gt or equivalently, that PE explog n.
E holds almost surely if PE .
Denition Matching. We say that two complex random variables ,
match
to order k for some integer k if one has ERe
m
Im
l
ERe
m
Im
l
for
all m, l with m l k.
Our main result is
Theorem Four Moment Theorem. For suciently small c
gt and suciently
large C
gt C
would suce the following holds for every lt lt and
k . Let M
ij
ip,jn
and M
ij
ip,jn
be matrix ensembles
obeying condition C with the the indicated constant C
, and assume that for each
i, j that
ij
and
ij
match to order . Let W, W
be the associated covariance
matrices. Assume also that p/n y for some lt y .
Let G R
k
R be a smooth function obeying the derivative bounds
j
Gx n
c
for all j and x R
k
.
Then for any i
lt i
lt i
k
n, and for n suciently large depending on
, k, c
we have
EGn
i
W, . . . , n
i
k
W EGn
i
W
,...,n
i
k
W
n
c
.
If
ij
and
ij
only match to order rather than , the conclusion still holds
provided that one strengthens to
j
Gx n
jc
for all j and x R
k
and any c
gt , provided that c
is suciently small
depending on c
.
TERENCE TAO AND VAN VU
This is an analogue of , Theorem for covariance matrices, with the main
dierence being that the exponential decay condition from , Theorem is
dropped, being replaced instead by the high moment condition in C. The main
results of , can all be strengthened similarly, using the same argument. The
value C
is ad hoc, and we make no attempt to optimize this constant.
Remark . The reason that we restrict the eigenvalues to the bulk of the spectrum
p i p is to guarantee that the density function
MP
is bounded away
from zero. In view of the results in , we expect that the result extends to the
edge of the spectrum as well. In particular, in view of the results in , it is likely
that the hard edge asymptotics of Forrester can be extended to a wider class of
ensembles. We will pursue this issue elsewhere.
.. Applications. One can apply Theorem in a similar way as its counterpart
, Theorem in order to obtain universality results for large classes of random
matrices, usually with less than four moment assumption. Let us demonstrate this
through an example concerning the universality of the sine kernel.
Using the explicit formula , Nagao and Wadati established the following
result for the complex Wishart ensemble.
Theorem Sine kernel for Wishart ensemble. Let k be an integer, let
fR
k
C be a continuous function with compact support and symmetric with
respect to permutations, and let lt u lt we assume all these quantities are
independent of n. Assume that
p nO thus y , and that W is given by
the complex Wishart ensemble. Let
,...,
p
be the nontrivial eigenvalues of W.
Then the quantity
E
i,...,i
k
p
fp
MP,
u
i
,...,p
MP,
u
i
k
converges as n to
R
m
ft
,...,t
k
detKt
i
,t
j
i,jk
dt
. . . dt
k
where Kx, y
sinxy
xy
is the sine kernel.
Remark . The results in allowed f to be bounded measurable rather than
continuous, but when we consider discrete ensembles later, it will be important to
keep f continuous.
Returning to the bulk, the following extension was established by Ben Arous and
Peche, as a variant of Johanssons result for random hermitian matrices.
We say that a complex random variable of mean zero and variance one is Gauss
divisible if has the same distribution as t
/
t
/
for some lt t lt
and some independent random variables
,
of mean zero and variance , with
distributed according to the complex gaussian.
See Section . for the asymptotic notation we will be using.
UNIVERSALITY FOR COVARIANCE MATRICES
Theorem Sine kernel for Gaussian divisible ensemble. Theorem which
is for the Wishart ensemble and for p nO can be extended to the case when
p n On
/
so y is still , and when M is an iid matrix obeying condition
C with C
, and with the
ij
gauss divisible.
Using Theorem and Theorem in exactly the same way we used , Theorem
and Johanssons theorem to establish , Theorem , we obtain the
following
Theorem Sine kernel for more general ensembles. Theorem can be extended
to the case when p n On
/
so y is still , and when M is an iid matrix
obeying condition C with C
suciently large C
would suce, and with
ij
have support on at least three points.
Remark . It was shown in , Corollary that if the real and imaginary parts
of a complex random variable were independent with mean zero and variance one,
and both were supported on at least three points, then matched to order with
a gauss divisible random variable with nite C
moment indeed, if one inspects
the convexity argument used to solve the moment problem in , Lemma , the
gauss divisible random variable could be taken to be the sum of a gaussian variable
and a discrete variable, and in particular is thus exponentially decaying.
The arguments in this paper will be a nonsymmetric version of those in .
Thus, for instance, everywhere eigenvectors are used in , left and right singular
vectors will be used instead and while in one often removes the last row and
column of a Hermitian n n matrix to make a n n submatrix, we will
instead remove just the last row of a p n matrix to form a p n matrix.
One way to connect the singular value problem for iid matrices to eigenvalue
problems for Wigner matrices is to form the augmented matrix
M
M
M
which is a pnpn Hermitian matrix, which has eigenvalues
M, . . . ,
p
M
together with n p eigenvalues at zero. Thus one can view the singular values of
an iid matrix as being essentially given by the eigenvalues of a slightly larger Her
mitian matrix which is of Wigner type except that the entries have been zeroed
out on two diagonal blocks. We will take advantage of thus augmented perspective
in some parts of the paper particularly when we wish to import results from
as black boxes, but in other parts it will in fact be more convenient to work with
M directly. In particular, the fact that many of the entries in are zero and
in particular, have zero mean and variance seems to make it dicult to apply
parts of the arguments particularly those that are probabilistic in nature, rather
than deterministic in directly to this matrix. Nevertheless one can view this
connection as a heuristic explanation as to why so much of the machinery in the
Hermitian eigenvalue problem can be transferred to the nonHermitian singular
value problem.
TERENCE TAO AND VAN VU
.. Extensions. In a very recent work, Erd os, Schlein, Yau and Yin ex
tended
Theorem to a large class of matrices, assuming that the distribution of
the entries
ij
is suciently smooth. While their results do not apply for entries
with discrete distributions, it allows one to extend Theorem to the case when t
is a negative power of n. Given this, one can use the argument in to remove
the requirement that the real and imaginary parts of
ij
be supported on at least
three points.
We can also have the following analogue of , Theorem .
Theorem Universality of averaged correlation function. Fix gt and u such
that lt u lt u lt . Let k and let f R
k
R be a continuous, compactly
supported function, and let W W
n,n
be a random covariance matrix. Then the
quantity
u
u
R
k
f
,...,
k
MP
u
k
p
k
n
u
n
MP
u
,...,u
k
n
MP
u
d
...d
k
du
converges as n to
R
k
f
,...,
k
detK
i
,
j
k
i,j
d
...d
k
,
where Kx, y is the Dyson sine kernel
Kx, y
sinx y
xy
.
The details are more or less the same as in and omitted.
.. Acknowledgments. We thank HorngTzer Yau for references.
. The gap property and the exponential decay removing trick
The following property plays an important role in .
Denition Gap property. Let M be a matrix ensemble obeying condition
C. We say that M obeys the gap property if for every , c gt independent of
n, and for every p i p, one has
i
W
i
Wn
c
with high
probability. The implied constants in this statement can depend of course on
and c.
As an analogue of , Theorem , we prove the following theorem, using the
same method with some modications.
Theorem Gap theorem. Let M
ij
ip,jn
obey condition C for
some C
, and suppose that the coecients
ij
are exponentially decaying in the
sense that P
ij
t
C
expt for all t C
for all i, j and some constants
C, C
gt . Then M obeys the gap property.
Even more recently, a similar result was also established by Peche.
UNIVERSALITY FOR COVARIANCE MATRICES
Next, we have the following analogue of , Theorem .
Theorem Four Moment Theorem with Gap assumption. For suciently
small c
gt and suciently large C
gt C
would suce the fol
lowing holds for every lt lt and k . Let M
ij
ip,jn
and
M
ij
ip,jn
be matrix ensembles obeying condition C with the indi
cated constant C
, and assume that for each i, j that
ij
and
ij
match to order
. Let W, W
be the associated covariance matrices. Assume also that M and M
obeys the gap property, and that p/n y for some lt y .
Let G R
k
R be a smooth function obeying the derivative bounds
j
Gx n
c
for all j and x R
k
.
Then for any i
lt i
lt i
k
n, and for n suciently large depending on
, k, c
we have
EGn
i
W, . . . , n
i
k
W EGn
i
W
,...,n
i
k
W
n
c
.
If
ij
and
ij
only match to order rather than , the conclusion still holds
provided that one strengthens to
j
Gx n
jc
for all j and x R
k
and any c
gt , provided that c
is suciently small
depending on c
.
This theorem is weaker than Theorem , as we assume the gap property. The
dierence comparing to , Theorem is that in the latter we assume exponential
decay rather than the gap property. However, this dierence is only a formality,
since in the proof of , Theorem , the only place we used exponential decay is
to prove the gap property via Theorem , Theorem .
The new step that enables us to remove the gap property altogether is the following
theorem, which asserts that the gap property is already guaranteed by condition
C, given C
bounded third moment.
Theorem Gap theorem. Assume that M
ij
ip,jn
satises condi
tion C with C
. Then M obeys the gap property.
Theorem follows directly from Theorems and .
The core of the proof of Theorem is Theorem , which allows us to insert
information such as the gap property into the test function G. We will also use this
theorem, combining with Theorem and Lemma , to prove Theorem .
The rest of the paper is organized as follows. The next three sections are devoted
to technical lemmas. The proofs of Theorems and are presented in Section
, assuming Theorems and . The proofs of these two theorems are presented
in Sections and , respectively.
TERENCE TAO AND VAN VU
. The main technical lemmas
Important note. The arguments in this paper are very similar to, and draw heavily
from, the previous paper of the authors. We recommend therefore that the
reader be familiar with that paper rst, before reading the current one.
In the proof of the Four Moment Theorem as well as the Gap Theorem for
n n Wigner matrices in , a crucial ingredient was a variant of the Delocaliza
tion Theorem of Erd os, Schlein, and Yau , , . This result asserts assuming
uniformly exponentially decaying distribution for the coecients that with over
whelming probability, all the unit eigenvectors of the Wigner matrix have coe
cients On
/o
thus the
energy of the eigenvector is spread out more
or less uniformly amongst the n coecients. When one just assumes uniformly
bounded C
moment rather than uniform exponential decay, the bound becomes
On
/O/C
instead where the implied constant in the exponent is uniform
in C
, of course.
Similarly, to prove the Four Moment and Gap Theorems in this paper, we will need
a Delocalization theorem for the singular vectors of the matrix M. We dene a
right singular vector u
i
resp. left singular vector v
i
with singular value
i
M
n
i
W
/
to be an eigenvector of W
n
M
M resp.
W
n
MM
with
eigenvalue
i
. Observe from the singular value decomposition that one can nd
orthonormal bases u
,...,u
p
C
n
and v
,...,v
p
C
p
for the corange
kerM
of M and of C
p
respectively, such that
Mu
i
i
v
i
and
M
v
i
i
u
i
.
In the generic case when the singular values are simple i.e. lt
lt . . .
p
, the
unit singular vectors u
i
,v
i
are determined up to multiplication by a complex phase
e
i
.
We will establish the following Erd osSchleinYau type delocalization theorem
analogous to , Proposition , which is an essential ingredient to Theorems
, and is also of some independent interest
Theorem Delocalization theorem. Suppose that p/n y for some lt y ,
and let M obey condition C for some C
. Suppose further that that
ij
K
almost surely for some K gt which can depend on n and all i, j, and that the
probability distribution of M is continuous. Let gt be independent of n. Then
with overwhelming probability, all the unit left and right singular vectors of M with
eigenvalue
i
in the interval a, b with a, b dened in have all coecients
uniformly of size OKn
/
log
n.
Strictly speaking, u
, . . . , up will span a slightly larger space than the corange if some of the
singular values
, . . . , p vanish. However, we shall primarily restrict attention to the generic
case in which this vanishing does not occur.
UNIVERSALITY FOR COVARIANCE MATRICES
The factors K log
n can probably be improved slightly, but anything which is
polynomial in K and log n will suce for our purposes. Observe that if M obeys
condition C, then each event
ij
K with K n
/C
say occurs with prob
ability On
. Thus in practice, we will be able to apply the above theorem
with K n
/C
without diculty. The continuity hypothesis is a technical one,
imposed so that the singular values are almost surely simple, but in practice we will
be able to eliminate this hypothesis by a limiting argument as none of the bounds
will depend on any quantitative measure of this continuity.
As with other proofs of delocalization theorems in the literature, Theorem is
in turn deduced from the following eigenvalue concentration bound analogous to
, Proposition
Theorem Eigenvalue concentration theorem. Let the hypotheses be as in
Theorem , and let gt be independent of n. Then for any interval I a, b
of length I K
log
n/n, one has with overwhelming probability uniformly
in I that
N
I
p
I
MP,y
x dx p,
where
N
I
ip
i
WI
is the number of eigenvalues in I.
We remark that a very similar result with slightly dierent hypotheses on the
parameters and on the underlying random variable distributions was recently es
tablished in , Corollary ..
We isolate one particular consequence of Theorem also established in
Corollary Concentration of the bulk. Let the hypotheses be as in Theorem .
Then there exists
gt independent of n such that with overwhelming probability,
one has a
i
Wb
for all p i p.
Proof. From Theorem , we see with overwhelming probability that the number
of eigenvalues in a
,b
is at least p, if
is suciently small depending
on . The claim follows.
.. Notation. Throughout this paper, n will be an asymptotic parameter going
to innity. Some quantities e.g. , y and C
will remain other independent of n,
while other quantities e.g. p, or the matrix M will depend on n. All statements
here are understood to hold only in the asymptotic regime when n is suciently
large depending on all quantities that are independent of n. We write X OY ,
Y X, X Y , or Y X if one has X CY for all suciently large
n and some C independent of n. Note however that C is allowed to depend on
other quantities independent of n, such as and y, unless otherwise stated. We
write X oY if X cnY where cn as n . We write X Y
TERENCE TAO AND VAN VU
or X Y if X Y X, thus for instance if p/n y for some lt y then
p n.
We write
for the complex imaginary unit, in order to free up the letter i to
denote an integer usually between and n.
We write X for the length of a vector X, A A
op
for the operator norm of
a matrix A, and A
F
trAA
/
for the Frobenius or HilbertSchmidt norm.
. Basic tools
.. Tools from linear algebra. In this section we recall some basic identities
and inequalities from linear algebra which will be used in this paper.
We begin with the Cauchy interlacing law and the Weyl inequalities
Lemma Cauchy interlacing law. Let p n.
i If A
n
is an n n Hermitian matrix, and A
n
is an n n minor,
then
i
A
n
i
A
n
i
A
n
for all i lt n.
ii If M
n,p
is a pn matrix, and M
n,p
is an pn minor, then
i
M
n,p
i
M
n,p
i
M
n,p
for all i lt p.
iii If p lt n, if M
n,p
is a p n matrix, and M
n,p
is a p n minor,
then
i
M
n,p
i
M
n,p
i
M
n,p
for all i p, with the
understanding that
M
n,p
. For p n, one can of course use the
transpose of ii instead.
Proof. Claim i follows from the minimax formula
i
A
n
inf
V dimV i
sup
vV v
v
A
n
v
where V ranges over idimensional subspaces in C
n
. Similarly, ii and iii follow
from the minimax formula
i
M
n,p
inf
V dimV inp
sup
vV v
M
n,p
v.
Lemma Weyl inequality. Let p n.
If A, B are n n Hermitian matrices, then
i
A
i
BAB
op
for all i n.
If M, N are p n matrices, then
i
M
i
NMN
op
for all
i p.
Proof. This follows from the same minimax formulae used to establish Lemma
.
UNIVERSALITY FOR COVARIANCE MATRICES
Remark . One can also deduce the singular value versions of Lemmas ,
from their Hermitian counterparts by using the augmented matrices . We omit
the details.
We have the following elementary formula for a component of an eigenvector of a
Hermitian matrix, in terms of the eigenvalues and eigenvectors of a minor
Lemma Formula for coordinate of an eigenvector. Let
A
n
A
n
X
X
a
be a nn Hermitian matrix for some a R and X C
n
, and let
v
x
be a unit
eigenvector of A
n
with eigenvalue
i
A
n
, where x C and v C
n
. Suppose
that none of the eigenvalues of A
n
are equal to
i
A
n
. Then
x
n
j
j
A
n
i
A
n
u
j
A
n
X
,
where u
A
n
,...,u
n
A
n
C
n
is an orthonormal eigenbasis correspond
ing to the eigenvalues
A
n
,...,
n
A
n
of A
n
.
Proof. See e.g. , Lemma .
This implies an analogous formula for singular vectors
Corollary Formula for coordinate of a singular vector. Let p, n , and let
M
p,n
M
p,n
X
be a p n matrix for some X C
p
, and let
u
x
be a right unit singular vector of
M
p,n
with singular value
i
M
p,n
, where x C and u C
n
. Suppose that none
of the singular values of M
p,n
are equal to
i
M
p,n
. Then
x
minp,n
j
j Mp,n
jMp,n
iMp,n
v
j
M
p,n
X
,
where v
M
p,n
,...,v
minp,n
M
p,n
C
p
is an orthonormal system of left
singular vectors corresponding to the nontrivial singular values of M
p,n
.
In a similar vein, if
M
p,n
M
p,n
Y
for some Y C
n
, and
vy
is a left unit singular vector of M
p,n
with singular
value
i
M
p,n
, where y C and v C
p
, and none of the singular values of
M
p,n
are equal to
i
M
p,n
, then
y
minp,n
j
j Mp,n
jMp,n
iMp,n
u
j
M
p,n
Y
,
TERENCE TAO AND VAN VU
where u
M
p,n
,...,u
minp,n
M
p,n
C
n
is an orthonormal system of right
singular vectors corresponding to the nontrivial singular values of M
p,n
.
Proof. We just prove the rst claim, as the second is proven analogously or by
taking adjoints. Observe that
u
x
is a unit eigenvector of the matrix
M
p,n
M
p,n
M
p,n
M
p,n
M
p,n
X
X
M
p,n
X
with eigenvalue
i
M
p,n
. Applying Lemma , we obtain
x
n
j
j
M
p,n
M
p,n
i
M
p,n
u
j
M
p,n
M
p,n
M
p,n
X
.
But u
j
M
p,n
M
p,n
M
p,n
j
M
p,n
v
j
M
p,n
for the minp, n
nontrivial singular values possibly after relabeling the j, and vanishes for trivial
ones, and
j
M
p,n
M
p,n
j
M
p,n
, so the claim follows.
The Stieltjes transform sz of a Hermitian matrix W is dened for complex z by
the formula
sz
n
n
i
i
Wz
.
It has the following alternate representation see e.g. , Chapter
Lemma . Let W
ij
i,jn
be a Hermitian matrix, and let z be a complex
number not in the spectrum of W. Then we have
s
n
z
n
n
k
kk
za
k
W
k
zI
a
k
where W
k
is the n n matrix with the k
th
row and column removed, and
a
k
C
n
is the k
th
column of W with the k
th
entry removed.
Proof. By Schurs complement,
kk
za
k
W
k
zI
a
k
is the k
th
diagonal entry of
W zI
. Taking traces, one obtains the claim.
.. Tools from probability theory. We will rely frequently on the following
concentration of measure result for projections of random vectors
Lemma Distance between a random vector and a subspace. Let X
,...,
n
C
n
be a random vector whose entries are independent with mean zero, variance
, and are bounded in magnitude by K almost surely for some K, where K
E
. Let H be a subspace of dimension d and
H
the orthogonal projec
tion onto H. Then
P
H
X
d t exp
t
K
.
In particular, one has
H
X
d OK log n
UNIVERSALITY FOR COVARIANCE MATRICES
with overwhelming probability.
Proof. See , Lemma the proof is a short application of Talagrands inequality
.
. Delocalization
The purpose of this section is to establish Theorem and Theorem . The
material here is closely analogous to , Sections ., ., as well as that of the
original results in , , and can be read independently of the other sections of
the paper. The recent paper also contains arguments and results closely related
to those in this section.
.. Deduction of Theorem from Theorem . We begin by showing how
Theorem follows from Theorem . We shall just establish the claim for the
right singular vectors u
i
, as the claim for the left singular vectors is similar. We x
and allow all implied constants to depend on and y. We can also assume that
K
log
n on as the claim is trivial otherwise.
As M is continuous, we see that the nontrivial singular values are almost surely
simple and positive, so that the singular vectors u
i
are well dened up to unit
phases. Fix i p it suces by the union bound and symmetry to show that
the event that
i
falls outside a , b or that the n
th
coordinate x of u
i
is
OKn
/
log
n holds with uniformly overwhelming probability.
Applying Corollary , it suces to show that with uniformly overwhelming prob
ability, either
i
a , b , or
minp,n
j
j
M
p,n
j
M
p,n
i
M
p,n
v
j
M
p,n
X
n
K
log
n
,
where M
M
p,n
X
. But if
i
a , b , then by
Theorem , one
can nd with uniformly overwhelming probability a set J , . . . , minp, n
with J K
log
n such that
j
M
p,n
i
M
p,n
OK
log
n/n for all
j J since
i
n
i
, we conclude that
j
M
p,n
i
M
p,n
OK
log
n.
In particular,
j
M
p,n
n. By Pythagoras theorem, the lefthand side of
is then bounded from below by
n
H
X
K
log
n
where H C
p
is the span of the v
j
M
p,n
for j J. But from Lemma and
the fact that X is independent of M
p,n
, one has
H
X
K
log
n
In the case p n, one would have to replace M
p,n
by its transpose to return to the regime
p n.
TERENCE TAO AND VAN VU
with uniformly overwhelming probability, and the claim follows.
It thus remains to establish Theorem .
.. A crude upper bound. Let the hypotheses be as in Theorem . We rst
establish a crude upper bound, which illustrates the techniques used to prove The
orem , and also plays an important direct role in that proof
Proposition Eigenvalue upper bound. Let the hypotheses be as in Theorem
. then for any interval I a , b of length I K log
n/n, one has with
overwhelming probability uniformly in I that
N
I
nI
where I denotes the length of I, and N
I
was dened in .
To prove this proposition, we suppose for contradiction that
N
I
CnI
for some large constant C to be chosen later. We will show that for C large enough,
this leads to a contradiction with overwhelming probability.
We follow the standard approach see e.g. of controlling the eigenvalue count
ing function N
I
via the Stieltjes transform
sz
p
p
j
j
Wz
.
Fix I. If x is the midpoint of I, I/, and z x
, we see that
Imsz
N
I
p
recall that p n so from one has
Imsz C.
Applying Lemma , with W replaced by the p p matrix
W
n
MM
which
only has the nontrivial eigenvalues, we see that
sz
p
p
k
kk
za
k
W
k
zI
a
k
,
where
kk
is the kk entry of W
,W
k
is the p p matrix with the k
th
row
and column of
W removed, and a
k
C
p
is the k
th
column of
W with the k
th
entry removed.
Using the crude bound Im
z
Imz
and , one concludes
p
p
k
Ima
k
W
k
zI
a
k
C.
UNIVERSALITY FOR COVARIANCE MATRICES
By the pigeonhole principle, there exists k p such that
Ima
k
W
k
zI
a
k
C.
The fact that k varies will cost us a factor of p in our probability estimates, but this
will not be of concern since all of our claims will hold with overwhelming probability.
Fix k. Note that
a
k
n
M
k
X
k
and
W
k
n
M
k
M
k
where X
k
C
n
is the adjoint of the k
th
row of M, and M
k
is the p n
matrix formed by removing that row. Thus, if we let v
M
k
,...,v
p
M
k
C
p
and u
M
k
,...,u
p
M
k
C
n
be coupled orthonormal systems of left and right
singular vectors of M
k
, and let
j
W
k
n
j
M
k
for j p be the
associated eigenvectors, one has
a
k
W
k
zI
a
k
p
j
a
k
v
j
M
k
j
W
k
z
.
and thus
Ima
k
W
k
zI
a
k
p
j
a
k
v
j
M
k
j
W
k
x
.
We conclude that
p
j
a
k
v
j
M
k
j
W
k
x
C
.
The expression a
k
v
j
M
k
can be rewritten much more favorably using as
a
k
v
j
M
k
j
M
k
n
X
k
u
j
M
k
.
The advantage of this latter formulation is that the random variables X
k
and
u
j
M
k
are independent for xed k.
Next, note that from and the Cauchy interlacing law Lemma one can
nd an interval J , . . . , p of length
J Cn
such that
j
W
k
I. We conclude that
jJ
j
M
k
n
X
k
u
j
M
k
C
.
Since
j
W
k
I, one has
j
M
k
n, and thus
jJ
X
k
u
j
M
k
n
C
.
TERENCE TAO AND VAN VU
The lefthand side can be rewritten using Pythagoras theorem as
H
X
k
, where
H is the span of the eigenvectors u
j
M
k
for j J. But from Lemma and
we see that this quantity is n with overwhelming probability, giving the desired
contradiction with overwhelming property even after taking the union bound in
k. This concludes the proof of Proposition .
.. Reduction to a Stieltjes transform bound. We now begin the proof of
Theorem in earnest. We continue to allow all implied constants to depend on
and y.
It suces by a limiting argument using Lemma to establish the claim under
the assumption that the distribution of M is continuous our arguments will not
use any quantitative estimates on this continuity.
The strategy is to compare s with the MarchenkoPastur Stieltjes transform
s
MP,y
z
R
MP,y
x
xz
dx.
A routine application of and the Cauchy integral formula yields the explicit
formula
s
MP,y
z
yz
yz
yz
yz
where we use the branch of
yz
yz with cut at a, b that is asymptotic
to yz as z . To put it another way, for z in the upper halfplane, s
MP,y
z
is the unique solution to the equation
s
MP,y
y z yzs
MP,y
z
with Ims
MP,y
z gt . Details of these computations can also be found in .
We have the following standard relation between convergence of Stieltjes transform
and convergence of the counting function
Lemma Control of Stieltjes transform implies control on ESD. Let /
/n, and L, , gt . Suppose that one has the bound
s
MP,y
z sz
with uniformly overwhelming probability for each z with Rez L and Imz
. Then for any interval I in a , b with I max,
log
, one has
N
I
n
I
MP,y
x dx nI
with overwhelming probability.
Proof. This follows from , Lemma strictly speaking, that lemma was phrased
for the semicircular distribution rather than the MarchenkoPastur distribution,
but an inspection of the proof shows the proof can be modied without diculty.
See also and , Corollary . for closely related lemmas.
UNIVERSALITY FOR COVARIANCE MATRICES
In view of this lemma, we see that to show Theorem , it suces to show that for
each complex number z with a/ Rez b/ and Imz
K
log
n
n
,
one has
sz s
MP,y
zo
with uniformly overwhelming probability.
For this, we return to the formula inserting the identities , one has
sz
p
p
k
kk
zY
k
where
Y
k
p
j
j
M
k
n
X
k
u
j
M
k
j
W
k
z
.
Suppose we condition M
k
and thus W
k
to be xed the entries of X
k
remain
independent with mean zero and variance , and thus since the u
j
are unit vectors
EY
k
M
k
p
j
j
M
k
n
j
W
k
z
p
n
zs
k
z
where
s
k
z
p
p
j
j
W
k
z
is the Stieltjes transform of W
k
.
From the Cauchy interlacing law Lemma we see that the dierence
sz
p
p
s
k
z
p
p
j
j
Wz
p
j
j
W
k
z
is bounded in magnitude by O
p
times the total variation of the function
z
on , , which is O
. Thus
p
p
s
k
z sz O
p
and thus
EY
k
M
k
p
n
p
n
zsz O
n
y o y ozsz
since p/n y o and / on.
We will shortly show a similar bound for Y
k
itself
Lemma Concentration of Y
k
. For each k p, one has Y
k
yo
y ozsz with overwhelming probability uniformly in k and I.
TERENCE TAO AND VAN VU
Meanwhile, we have
kk
n
X
k
and hence by Lemma ,
kk
o with overwhelming probability again
uniformly in k and I. Inserting these bounds into , one obtains
sz
p
p
k
z y o y ozsz
with overwhelming probability thus sz almost solves in some sense. From
the quadratic formula, the two solutions of are s
MP,y
z and
yz
yz
s
MP,y
z. One concludes that with overwhelming probability, one has either
sz s
MP,y
zo
or
sz
yz
yz
o
or
sz
yz
yz
s
MP,y
zo
with the convention that
yz
yz
when y . By using a n
net say
of possible zs and using the union bound and the fact that sz has a Lipschitz
constant of at most On
say in the region of interest we may assume that the
above trichotomy holds for all z with a/ Rez b/ and Imz n
say.
When Imz n
, then sz, s
MP,y
z are both o, and so holds in this
case. By continuity, we thus claim that either holds for all z in the domain of
interest, or there exists a z such that as well as one of or both hold.
From one has
s
MP,y
z
yz
yz
s
MP,y
z
yz
which implies that the separation between s
MP,y
z from
yz
yz
is bounded from
below, which implies that , cannot both hold for n large enough. Simi
larly, from we see that
yz
yz
s
MP,y
z
yz
yz
yz
since y z
yz has zeroes only when z a, b, and z is bounded away from
these singularities, we see also that , cannot both hold for n large enough.
Thus the continuity argument shows that holds with uniformly overwhelming
probability for all z in the region of interest for n large enough, which gives
and thus Theorem .
UNIVERSALITY FOR COVARIANCE MATRICES
. Proof of Theorem and Theorem
We rst prove Theorem . The arguments follow those in .
We begin by observing from Markovs inequality and the union bound that one has
ij
,
ij
n
/C
say for all i, j with probability On
. Thus, by truncation
and adjusting the moments appropriately, using Lemma to absorb the error,
one may assume without loss of generality that
ij
,
ij
n
/C
almost surely for all i, j. Next, by a further approximation argument we may
assume that the distribution of M, M
is continuous. This is a purely qualitative
assumption, to ensure that the singular values are almost surely simple our bounds
will not depend on any quantitative measure on the continuity, and so the general
case then follows by a limiting argument using Lemma .
The key technical step is the following theorem, whose proof is delayed to the next
section.
Theorem Truncated Four Moment Theorem. For suciently small c
gt
and suciently large C
gt the following holds for every lt lt and k .
Let M
ij
ip,jn
and M
ij
ip,jn
be matrix ensembles obeying
condition C for some C
, as well as . Assume that p/n y for some lt
y , and that
ij
and
ij
match to order .
Let G R
k
R
k
R be a smooth function obeying the derivative bounds
j
Gx
,...,x
k
,q
,...,q
k
n
c
for all j and x
,...,x
k
R, q
,...,q
k
R, and such that G is supported
on the region q
,...,q
k
n
c
, and the gradient is in all k variables.
Then for any i
lt i
lt i
k
n, and for n suciently large depending on
, k, c
we have
EG
n
i
M, . . . ,
n
i
k
M, Q
i
M, . . . , Q
i
k
M
EG
n
i
M
,...,
n
i
k
M
,Q
i
M
,...,Q
i
k
M
n
c
.
If
ij
,
ij
match to order , then the conclusion still holds as long as one strengthens
to
j
Gx
,...,x
k
,q
,...,q
k
n
jc
for some c
gt , if c
is suciently small depending on c
.
Given a p n matrix M we form the augmented matrix M dened in , whose
eigenvalues are
M, . . . ,
p
M, together with the eigenvalue with multi
plicity n p if p lt n. For each i p, we introduce in analogy with the
TERENCE TAO AND VAN VU
arguments in the quantities
Q
i
M
iM
n
i
M
n
jpji
j
M
i
M
np
i
M
p
j
j
M
i
M
.
The factor of
n
in Q
i
M is present to align the notation here with that in ,
in which one dilated the matrix by
n. We set Q
i
M if the singular value
i
is repeated, but this event occurs with probability zero since we are assuming
M to be continuously distributed.
The gap property on M ensures an upper bound on Q
i
M
Lemma . If M satises the gap property, then for any c
gt independent of
n, and any p i p, one has Q
i
Mn
c
with high probability.
Proof. Observe the upper bound
Q
i
M
n
jpji
j
M
i
M
np
n
i
M
.
From Corollary we see that with overwhelming probability,
i
M
/n is bounded
away from zero, and so
np
niM
O/n. To bound the other term in , one
repeats the proof of , Lemma .
By applyign a truncation argument exactly as in , Section ., one can now
remove the hypothesis in Theorem that G is supported in the region q
,...,q
k
n
c
. In particular, one can now handle the case when G is independent of q
,...,q
k
and Theorem follows after making the change of variables
n
and using
the chain rule and Corollary .
Next, we prove Theorem , assuming both Theorems and . The main
observation here is the following lemma.
Lemma Matching lemma. Let be a complex random variable with mean
zero, unit variance, and third moment bounded by some constant a. Then there
exists a complex random variable
with support bounded by the ball of radius O
a
centered at the origin and in particular, obeying the exponential decay hypothesis
uniformly in for xed a which matches to third order.
Proof. In order for
to match to third order, it suces that
have mean zero,
variance , and that E
E
and E
E
.
Accordingly, let C
be the set of pairs E
,E
where
ranges over
complex random variables with mean zero, variance one, and compact support.
Clearly is convex. It is also invariant under the symmetry z, w e
i
z, e
i
w
UNIVERSALITY FOR COVARIANCE MATRICES
for any phase . Thus, if z, w , then z, e
i/
w , and hence by convexity
,
e
i/
w , and hence by convexity and rotation invariance , w
whenever w
w. Since z, w and ,
w both lie in , by convexity
cz, lies in it also for some absolute constant c gt , and so again by convexity
and rotation invariance z
, whenever z
cz. One last application of
convexity then gives z
/, w
/ whenever z
cz and w
w.
It is easy to construct complex random variables with mean zero, variance one,
compact support, and arbitrarily large third moment. Since the third moment
is comparable to z w, we thus conclude that contains all of C
, i.e. every
complex random variable with nite third moment with mean zero and unit variance
can be matched to third order by a variable of compact support. An inspection of
the argument shows that if the third moment is bounded by a then the support
can also be bounded by O
a
.
Now consider a random matrix M as in Theorem with atom variables
ij
. By
the above lemma, for each i, j, we can nd
ij
which satises the exponential decay
hypothesis and match
ij
to third order. Let q be a smooth cuto to the region
qn
c
for some c
gt independent of n, and let p i p. By Theorem
,M
satises the gap property. By Lemma ,
EQ
i
M
On
c
for some c
gt independent of n, so by Theorem one has
EQ
i
M On
c
for some c
gt independent of n. We conclude that M also obeys the gap property.
The next two sections are devoted to the proofs of Theorem and Theorem ,
respectively.
Remark . The above trick to remove the exponential decay hypothesis for The
orem also works to remove the same hypothesis in , Theorem . The point
is that in the analogue of Theorem in that paper implicit in , Section .,
the exponential decay hypothesis is not used anywhere in the argument only a
uniformly bounded C
moment for C
large enough is required, as is the case here.
Because of this, one can replace all the exponential decay hypotheses in the results
of , by a hypothesis of bounded C
moment we omit the details.
. The proof of Theorem
It remains to prove Theorem . By telescoping series, it suces to establish a
bound
EG
n
i
M, . . . ,
n
i
k
M, Q
i
M, . . . , Q
i
k
M
EG
n
i
M
,...,
n
i
k
M
,Q
i
M
,...,Q
i
k
M
n
c
TERENCE TAO AND VAN VU
under the assumption that the coecients
ij
,
ij
of M and M
are identical except
in one entry, say the qr entry for some q p and r n, since the claim then
follows by interchanging each of the pn On
entries of M into M
separately.
Write Mz for the matrix M or M
with the qr entry replaced by z. We apply
the following proposition, which follows from a lengthy argument in
Proposition Replacement given a good conguration. Let the notation and
assumptions be as in Theorem . There exists a positive constant C
independent
of k such that the following holds. Let
gt . We condition i.e. freeze all the
entries of Mz to be constant, except for the qr entry, which is z. We assume that
for every j k and every z n
/
whose real and imaginary parts are
multiples of n
C
, we have
Singular value separation For any i n with i i
j
n
, we have
n
i
Mz
n
ij
Mz n
ii
j
.
Also, we assume
n
ij
Az n
n.
Delocalization at i
j
If u
ij
Mz C
n
,v
ij
Mz C
p
are unit right and
left singular vectors of Mz, then
e
q
v
ij
Mz, e
r
u
ij
Mz n
/
.
For every
P
ij,
Mze
q
,P
ij,
Mze
r
/
n
/
,
whenever P
ij,
resp. P
ij ,
is the orthogonal projection to the span of
right singular vectors u
i
Mz resp. left singular vectors v
i
Mz cor
responding to singular values
i
Az with
ii
j
lt
.
We say that M, e
q
,e
r
are a good conguration for i
,...,i
k
if the above proper
ties hold. Assuming this good conguration, then we have if
ij
and
ij
match
to order , or if they match to order and holds.
Proof. This follows by applying , Proposition to the p np n Hermitian
matrix Az
nMz, where Mz is the augmented matrix of Mz, dened
in . Note that the eigenvalues of Az are
n
Mz, . . . ,
n
p
Mz
and , and that the eigenvalues are given up to unit phases by
v
j
Mz
u
j
Mz
.
Note also that the analogue of in , Proposition is trivially true if
is
comparable to n, so one can restrict attention to the regime
on.
In view of the above proposition, we see that to conclude the proof of Theorem
and thus Theorem it suces to show that for any
gt , that M, e
q
,e
r
are a
good conguration for i
,...,i
k
with overwhelming probability, if C
is suciently
large depending on
cf. , Proposition .
UNIVERSALITY FOR COVARIANCE MATRICES
Our main tools for this are Theorem and Theorem . Actually, we need a
slight variant
Proposition . The conclusions of Theorem and Theorem continue to hold
if one replaces the qr entry of M by a deterministic number z On
/O/C
.
This is proven exactly as in , Corollary and is omitted.
We return to the task of establishing a good conguration with overwhelming prob
ability. By the union bound we may x j k, and also x the z n
/
whose real and imaginary parts are multiples of n
C
. By the union bound again
and Proposition , the eigenvalue separation condition holds with overwhelm
ing probability for every i n with i j n
if C
is suciently large, as
does . A similar argument using Pythagoras theorem and Corollary gives
with overwhelming probability noting as before that we may restrict atten
tion to the regime
on. Corollary also gives with overwhelming
probability. This gives the claim, and Theorem follows.
. Proof of Theorem
We now prove Theorem , closely following the analogous arguments in .
Using the exponential decay condition, we may truncate the
ij
and renormalise
moments, using Lemma to assume that
ij
log
O
n
almost surely. By a limiting argument we may assume that M has a continuous
distribution, so that the singular values are almost surely simple.
We write i
instead of i, p
instead of p, and write N
p
n. As in , the
strategy is to propagate a narrow gap for M M
p,n
backwards in the p variable,
until one can use Theorem to show that the gap occurs with small probability.
More precisely, for any i l lt i p p
, we let M
p,n
be the p n matrix
formed using the rst p rows of M
p,n
, and we dene following the regularized
gap
g
i,l,p
inf
iilltiip
N
i
M
p,n
N
i
M
p,n
mini
i
, log
C
N
log
.
N
,
where C
gt is a large constant to be chosen later. It will suce to show that
g
i,,p
n
c
.
The main tool for this is
Lemma Backwards propagation of gap. Suppose that p
/ p lt p
and
l p/ is such that
g
i,l,p
TERENCE TAO AND VAN VU
for some lt which can depend on n, and that
g
i,l,p
m
g
i,l,p
for some m with
m
/
.
Let X
p
be the p
th
row of M
p,n
, and let u
M
p,n
,...,u
p
M
p,n
be an orthonor
mal system of right singular vectors of M
p,n
associated to
M
p,n
,...,
p
M
p,n
.
Then one of the following statements hold
i Macroscopic spectral concentration There exists i
lt i
p
with i
i
log
C/
n such that
n
i
M
p,n
n
i
M
p,n
/
explog
.
ni
i
.
ii Small inner products There exists p/ i
i
l lt i
i
/p with i
i
log
C/
n such that
ijlti
X
p
u
j
M
p,n
i
i
m/
log
.
n
.
iii Large singular value For some i p one has
i
M
p,n
nexplog
.
n
/
.
iv Large inner product in bulk There exists p/ i /p such
that
X
p
u
i
M
p,n
explog
.
n
/
.
v Large row We have
X
p
nexplog
.
n
/
.
vi Large inner product near i
There exists p/ i /p with
ii
log
C
n such that
X
p
u
i
M
p,n
m/
nlog
.
n.
Proof. This follows by applying
, Lemma to the p n p n
Hermitian matrix
A
pn
n
M
p,n
M
p,n
,
which after removing the bottom row and rightmost column which is X
p
, plus
p zeroes yields the p n p n Hermitian matrix
A
pn
n
M
p,n
M
p,n
Strictly speaking, there are some harmless adjustments by constant factors that need to be
made to this lemma, ultimately coming from the fact that n, p, n p are only comparable up to
constants, rather than equal, but these adjustments make only a negligible change to the
proof of
that lemma.
UNIVERSALITY FOR COVARIANCE MATRICES
which has eigenvalues
n
M
p,n
,...,
n
p
M
p,n
and , and an orthonor
mal eigenbasis that includes the vectors
u
j
M
p,n
v
j
M
p,n
for j p. The large
coecient event in , Lemma iii cannot occur here, as A
pn
has zero
diagonal.
By repeating the arguments in , Section . almost verbatim, it then suces
to show that
Proposition Bad events are rare. Suppose that p
/ p lt p
and l p/,
and set n
for some suciently small xed gt . Then
a The events i, iii, iv, v in Lemma all fail with high probability.
b There is a constant C
such that all the coecients of the right singular
vectors u
j
M
p,n
for p/ j /p are of magnitude at most
n
/
log
C
n with overwhelming probability. Conditioning M
p,n
to be a
matrix with this property, the events ii and vi occur with a conditional
probability of at most
m
n
.
c Furthermore, there is a constant C
depending on C
,,C
such that if
lC
and M
p,n
is conditioned as in b, then ii and vi in fact occur
with a conditional probability of at most
m
log
C
nn
.
But Proposition can be proven by repeating the proof of , Proposition
with only cosmetic changes, the only signicant dierence being that Theorem
and Theorem are applied instead of , Theorem and , Proposition
respectively.
References
G. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices, book to be
published by Cambridge Univ. Press.
Z. D. Bai and J. Silverstein, Spectral analysis of large dimensional random matrices, Mathe
matics Monograph Series , Science Press, Beijing .
G. Ben Arous and S. Peche, Universality of local eigenvalue statistics for some sample
covari
ance matrices, Comm. Pure Appl. Math. , no. , .
K. Costello and V. Vu, Concentration of random determinants and permanent estimators, to
appear in SIAM discrete math.
P. Deift, Orthogonal polynomials and random matrices a RiemannHilbert approach.
Courant Lecture Notes in Mathematics, . New York University, Courant Institute of Math
ematical Sciences, New York American Mathematical Society, Providence, RI, .
P. Deift, Universality for mathematical and physical systems. International Congress of Math
ematicians Vol. I, , Eur. Math. Soc., Zrich, .
A. Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal.
Appl. , .
A. Edelman, The distribution and moments of the smallest eigenvalue of a random matrix of
Wishart type, Linear Algebra Appl. , .
L. Erdos, B. Schlein and HT. Yau, Semicircle law on short scales and delocalization of
eigenvectors for Wigner random matrices, to appear in Annals of Probability.
L. Erdos, B. Schlein and HT. Yau, Local semicircle law and complete delocalization for
Wigner random matrices, to appear in Comm. Math. Phys.
TERENCE TAO AND VAN VU
L. Erdos, B. Schlein and HT. Yau, Wegner estimate and level repulsion for Wigner random
matrices, submitted.
L. Erdos, J. Ramirez, B. Schlein and HT. Yau, Universality of sinekernel for Wigner matrices
with a small Gaussian perturbation, arXiv..
L. Erdos, J. Ramirez, B. Schlein and HT. Yau, Bulk universality for Wigner matrices,
arXiv.
L. Erdos, B. Schlein, HT. Yau and J. Yin, The local relaxation ow approach to universality
of the local statistics for random matrices, arXiv..
L. Erdos, J. Ramirez, B. Schlein, T. Tao, V. Vu, and HT. Yau, Bulk universality for Wigner
hermitian matrices with subexponential decay, to appear in Math. Res. Lett.
O. N. Feldheim and S. Sodin, A universality result for the smallest eigenvalues of certain
sample covariance matrices, preprint.
P. J. Forrester, The spectrum edge of random matrix ensembles, Nuclear Phys. B ,
.
P. J. Forrester, Exact results and universal asymptotics in the Laguerre random matrix
ensemble, J. Math. Phys. , no. , .
P. J. Forrester, N. S. Witte, The distribution of the rst eigenvalue spacing at the hard edge
of the Laguerre unitary ensemble, Kyushu J. Math. , no. , .
J. Ginibre, Statistical Ensembles of Complex, Quaternion, and Real Matrices, Journal of
Mathematical Physics , .
F. Gotze, A. Tikhomirov, Rate of convergence to the semicircular law, SFB Universitat
Bielefeld, Preprint , .
F. Gotze, A. Tikhomirov, Rate of convergence in probability to the MarchenkoPastur law,
Bernoulli , no. , .
A. Guionnet and O. Zeitouni, Concentration of the spectral measure for large matrices,
Electron. Comm. Probab. , electronic.
J. Gustavsson, Gaussian uctuations of eigenvalues in the GUE, Ann. Inst. H. Poincar
Probab. Statist. , no. , .
K. Johansson, Universality of the local spacing distribution in certain ensembles of Hermitian
Wigner matrices, Comm. Math. Phys. , no. , .
I. Johnstone, On the distribution of largest principal component, Ann. Statist. , .
N. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy. Amer
ican Mathematical Society Colloquium Publications, . American Mathematical Society,
Providence, RI, .
M. Ledoux, The concentration of measure phenomenon, Mathematical survey and mono
graphs, volume , AMS .
J. W. Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeit
srechnung, Math. Z. , .
V. A. Marchenko, L. A. Pastur, The distribution of eigenvalues in certain sets of random
matrices, Mat. Sb. , .
M.L. Mehta, Random Matrices and the Statistical Theory of Energy Levels, Academic Press,
New York, NY, .
M.L. Mehta and M. Gaudin, On the density of eigenvalues of a random matrix, Nuclear
Phys.
,.
T. Nagao, K. Slevin, Laguerre ensembles of random matrices nonuniversal correlation func
tions, J. Math. Phys. , no. , .
T. Nagao, M. Wadati, Correlation functions of random matrix ensembles related to classical
orthogonal polynomials, J. Phys. Soc. Japan , no. , .
V. Paulauskas and A. Rackauskas, Approximation theory in the central limit theorem, Kluwer
Academic Publishers, .
L. A. Pastur, Spectra of random selfadjoint operators. Russian Math. Surveys, ,
.
S. Peche, Universality in the bulk of the spectrum for complex sample covariance matrices,
preprint.
B. Rider, J. Ramirez, Diusion at the random matrix hard edge, to appear in Comm. Math.
Phys.
A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm.
Math. Phys. , no. , .
UNIVERSALITY FOR COVARIANCE MATRICES
A. Soshnikov, Gaussian limit for determinantal random point elds, Ann. Probab.
.
A. Soshnikov, A note on universality of the distribution of the largest eigenvalues in certain
sample covariance matrices, J. Statist. Phys. , no. , .
M. Talagrand, A new look at independence, Ann. Prob. , no. , .
T. Tao and V. Vu, On random matrices singularity and determinant, Random Structures
Algorithms , no. , .
T. Tao and V. Vu with an appendix by M. Krishnapur, Random matrices Universality of
ESDs and the circular law, to appear in Annals of Probability.
T. Tao and V. Vu, Random matrices The distribution of the smallest singular values Uni
verality at the Hard Edge, To appear in GAFA.
, T. Tao and V. Vu, Random matrices The local statistics of eigenvalues, to appear in Acta
Math..
T. Tao and V. Vu, Random matrices universality of local eigenvalue statistics up to the
edge, to appear in Comm. Math. Phys.
J. von Neumann and H. Goldstine, Numerical inverting matrices of high order, Bull. Amer.
Math. Soc. , , .
P. Wigner, On the distribution of the roots of certain symmetric matrices, The Annals of
Mathematics .
K. W. Wachter, Strong limits of random matrix spectra for sample covariance matrices of
independent elements, Ann. Probab., , .
Y. Q. Yin, Limiting spectral distribution for a class of random matrices. J. Multivariate
Anal., , .
Department of Mathematics, UCLA, Los Angeles CA
Email address taomath.ucla.edu
Department of Mathematics, Rutgers, Piscataway, NJ
Email address vanvumath.rutgers.edu
TERENCE TAO AND VAN VU
We often abbreviate i W as i . Remark . In this paper we will focus primarily on the case y ,
but several of our results extend to other values of y as well. The case p gt n can of course
be deduced from the p lt n case after some minor notational changes by transposing the
matrix M , which does not aect the nontrivial eigenvalues of the covariance matrix. One can
also easily normalise the variance of the entries to be some other / quantity than if one
wishes. Observe that the quantities i ni can be interpreted as the nontrivial singular values of
the original matrix M , and , . . . , p can also be interpreted as the eigenvalues of the p p
matrix n M M . It will be convenient to exploit all three of these spectral interpretations of , . . .
, p in this paper. Condition C is analogous to Condition C for Wignertype matrices in , but
with the exponential decay hypothesis relaxed to polynomial decay only. The wellknown
MarchenkoPastur law governs the bulk distribution of the eigenvalues , . . . , p of W Theorem
MarchenkoPastur law. Assume Condition C with C , and suppose that p/n y for some lt y .
Then for any x gt , the random variables i p i W x p x converge in probability to MP,y x dx,
where and When furthermore M is iid, one can also obtain the case C . Proof. For C , see ,
for C gt , see for the C iid case, see . Further results are known on the rate of convergence
see . In this paper we are concerned instead with the local eigenvalue statistics. A model
case is the complex Wishart ensemble, in which the ij are iid variables which are complex
gaussians with mean zero and variance . In this case, the distribution of the eigenvalues , . . .
, n of W can be explicitly computed as a special case of the Laguerre unitary ensemble. For
instance, when p n, the joint distribution is given by the density function
n
MP,y x
xy y
b xx aa,b x b y .
a
n , . . . , n cn
iltjn
i j exp
i/
i
for some explicit normalization constant cn. Very similarly to the GUE case, one can use this
explicit formula to directly compute several local statistics, including the distribution of the
largest and smallest eigenvalues , the correlation functions etc. Also in similarity to the GUE
We rst need some denitions. . the conclusion still holds provided that one strengthens to for
all j and x Rk and any c gt . . For some earlier results in this direction. we refer to . Denition
Frequent events. We say that two complex random variables . E holds asymptotically almost
surely if PE o.jn be matrix ensembles obeying condition C with the the indicated constant C .
that PE explog n. match to order k for some integer k if one has ERem Iml ERe m Im l for all
m. Assume also that p/n y for some lt y . Let E be an event depending on n. . as an analogue
of a recent result in . E holds with high probability if PE Onc for some constant c gt
independent of n. . nik W EGni W . k. . j Gx njc . Let M ij ip. . . E holds with overwhelming
probability if PE OC nC for every constant C gt or equivalently.. The Four Moment Theorem.
Denition Matching. and for n suciently large depending on . The goal of this paper is to
establish a Four Moment theorem for random covariance matrices. For suciently small c gt
and suciently large C gt C would suce the following holds for every lt lt and k . c we have
EGni W . .UNIVERSALITY FOR COVARIANCE MATRICES case. l with m l k. W be the
associated covariance matrices. . Let G Rk R be a smooth function obeying the derivative
bounds for all j and x Rk . If ij and ij only match to order rather than . and the references
therein.jn and M ij ip. it is widely conjectured that these statistics hold for a much larger class
of random matrices. . j that ij and ij match to order . and assume that for each i. Let W. j Gx
nc Then for any i lt i lt ik n. This theorem asserts that all local statistics of the eigenvalues of
Wn is determined by the rst four moments of the entries. nik W nc . Our main result is
Theorem Four Moment Theorem. provided that c is suciently small depending on c . . E
holds almost surely if PE .
usually with less than four moment assumption. Let . we expect that the result extends to the
edge of the spectrum as well. of mean zero and variance . with the main dierence being that
the exponential decay condition from . . Let us demonstrate this through an example
concerning the universality of the sine kernel. pMP. Remark .jk dt .. dtk Rm where Kx. ui .
and let lt u lt . Applications.ik p f pMP. e We say that a complex random variable of mean
zero and variance one is Gauss divisible if has the same distribution as t/ t/ for some lt t lt
and some independent random variables . Nagao and Wadati established the following result
for the complex Wishart ensemble. See Section . p be the nontrivial eigenvalues of W . it will
be important to keep f continuous. y sinxy xy is the sine kernel. can all be strengthened
similarly. . . in view of the results in . Theorem in order to obtain universality results for large
classes of random matrices. Using the explicit formula . Assume that p n O thus y . . with
distributed according to the complex gaussian. Returning to the bulk. for the asymptotic
notation we will be using. Remark . Let k be an integer.. Then the quantity E i . and that W is
given by the complex Wishart ensemble. using the same argument. we assume all these
quantities are independent of n. tk detKti .. TERENCE TAO AND VAN VU This is an
analogue of . . The reason that we restrict the eigenvalues to the bulk of the spectrum p i p is
to guarantee that the density function MP is bounded away from zero. The main results of .
The value C is ad hoc. . tj i. being replaced instead by the high moment condition in C. uik
converges as n to f t .. In particular. let f Rk C be a continuous function with compact support
and symmetric with respect to permutations. Theorem Sine kernel for Wishart ensemble. it is
likely that the hard edge asymptotics of Forrester can be extended to a wider class of
ensembles. . One can apply Theorem in a similar way as its counterpart . . and we make no
attempt to optimize this constant.. Theorem is dropped. . The results in allowed f to be
bounded measurable rather than continuous. as a variant of Johanssons result for random
hermitian matrices. . . . . the following extension was established by Ben Arous and Pech. but
when we consider discrete ensembles later. We will pursue this issue elsewhere. Theorem
for covariance matrices. . . In view of the results in . .
left and right singular vectors will be used instead. Nevertheless one can view this
connection as a heuristic explanation as to why so much of the machinery in the Hermitian
eigenvalue problem can be transferred to the nonHermitian singular value problem. which
has eigenvalues M . It was shown in . . . Theorem . Theorem can be extended to the case
when p n On/ so y is still . the fact that many of the entries in are zero and in particular. Using
Theorem and Theorem in exactly the same way we used . Thus. and both were supported on
at least three points. One way to connect the singular value problem for iid matrices to
eigenvalue problems for Wigner matrices is to form the augmented matrix M M M which is a
pnpn Hermitian matrix. the gauss divisible random variable could be taken to be the sum of a
gaussian variable and a discrete variable. everywhere eigenvectors are used in . The
arguments in this paper will be a nonsymmetric version of those in . We will take advantage
of thus augmented perspective in some parts of the paper particularly when we wish to
import results from as black boxes. if one inspects the convexity argument used to solve the
moment problem in . we obtain the following Theorem Sine kernel for more general
ensembles. and when M is an iid matrix obeying condition C with C suciently large C would
suce. Thus one can view the singular values of an iid matrix as being essentially given by the
eigenvalues of a slightly larger Hermitian matrix which is of Wigner type except that the
entries have been zeroed out on two diagonal blocks. then matched to order with a gauss
divisible random variable with nite C moment indeed. for instance. Theorem and Johanssons
theorem to establish . . In particular. rather than deterministic in directly to this matrix. we will
instead remove just the last row of a p n matrix to form a p n matrix. . and in particular is thus
exponentially decaying. Remark . and with ij have support on at least three points. . Corollary
that if the real and imaginary parts of a complex random variable were independent with
mean zero and variance one.UNIVERSALITY FOR COVARIANCE MATRICES Theorem
Sine kernel for Gaussian divisible ensemble. have zero mean and variance seems to make it
dicult to apply parts of the arguments particularly those that are probabilistic in nature. but in
other parts it will in fact be more convenient to work with M directly. and when M is an iid
matrix obeying condition C with C . and with the ij gauss divisible. Theorem which is for the
Wishart ensemble and for p n O can be extended to the case when p n On/ so y is still . p M
together with n p eigenvalues at zero. Lemma . and while in one often removes the last row
and column of a Hermitian n n matrix to make a n n submatrix.
We thank HorngTzer Yau for references. Fix gt and u such that lt u lt u lt .jn obey condition C
for some C . Extensions. Given this. . The gap property and the exponential decay removing
trick The following property plays an important role in . where Kx. . y is the Dyson sine kernel
Kx. In a very recent work. one can use the argument in to remove the requirement that the
real and imaginary parts of ij be supported on at least three points. a similar result was also
established by Pch. Then the quantity u k pk u . . using the same method with some
modications. Then M obeys the gap property. and suppose that the coecients ij are
exponentially decaying in the sense that Pij tC expt for all t C for all i. . and let W Wn.
Acknowledgments. x y The details are more or less the same as in and omitted. k detKi .
Theorem . .. one has i W i W nc with high probability. compactly supported function. Even
more recently. We say that M obeys the gap property if for every . j and some constants C.
Yau and Yin exo tended Theorem to a large class of matrices.. . u f . dk . it allows one to
extend Theorem to the case when t is a negative power of n. and for every p i p. Theorem
Universality of averaged correlation function. Theorem . TERENCE TAO AND VAN VU . dk
du k n u Rk MP u nMP u nMP u converges as n to Rk f . We can also have the following
analogue of . . Denition Gap property. . Schlein. Erds. . k d . . .n be a random covariance
matrix. While their results do not apply for entries with discrete distributions. j k i. . Theorem
Gap theorem. .j d . Let k and let f Rk R be a continuous. y sinx y . The implied constants in
this statement can depend of course on and c. . . assuming that the distribution of the entries
ij is suciently smooth. . C gt . . . e e . c gt independent of n. we prove the following theorem.
Let M be a matrix ensemble obeying condition C. As an analogue of . Let M ij ip.
nik W EGni W . Let W. and assume that for each i.UNIVERSALITY FOR COVARIANCE
MATRICES Next. Theorem is that in the latter we assume exponential decay rather than the
gap property. assuming Theorems and . The proofs of Theorems and are presented in
Section . this dierence is only a formality.jn satises condition C with C . Then M obeys the
gap property. Theorem .jn be matrix ensembles obeying condition C with the indi cated
constant C . to prove Theorem . j Gx njc Theorem Gap theorem. However. The new step that
enables us to remove the gap property altogether is the following theorem. . Assume that M ij
ip. j that ij and ij match to order . . Then for any i lt i lt ik n. W be the associated covariance
matrices.jn and M ij ip. the only place we used exponential decay is to prove the gap property
via Theorem . nik W nc . This theorem is weaker than Theorem . . and that p/n y for some lt y
. which allows us to insert information such as the gap property into the test function G. the
conclusion still holds provided that one strengthens to for all j and x Rk and any c gt . . The
core of the proof of Theorem is Theorem . c we have EGni W . Let G Rk R be a smooth
function obeying the derivative bounds j Gx nc for all j and x Rk . Theorem . . we have the
following analogue of . given C bounded third moment. since in the proof of . as we assume
the gap property. Let M ij ip. which asserts that the gap property is already guaranteed by
condition C. . and for n suciently large depending on . combining with Theorem and Lemma .
provided that c is suciently small depending on c . k. . Theorem follows directly from
Theorems and . If ij and ij only match to order rather than . . The proofs of these two
theorems are presented in Sections and . Theorem Four Moment Theorem with Gap
assumption. respectively. The rest of the paper is organized as follows. The dierence
comparing to . . We will also use this theorem. The next three sections are devoted to
technical lemmas. Theorem . Assume also that M and M obeys the gap property. For
suciently small c gt and suciently large C gt C would suce the following holds for every lt lt
and k .
. We recommend therefore that the reader be familiar with that paper rst. . . . In the generic
case when the singular values are simple i. . and Yau . The main technical lemmas Important
note. p . j. to prove the Four Moment and Gap Theorems in this paper. p vanish. . Suppose
that p/n y for some lt y . . and that the probability distribution of M is continuous. the previous
paper of the authors. u will span a slightly larger space than the corange if some of the p
singular values . Strictly speaking.e. . and draw heavily from. . However. b with a. left
singular vector vi with singular value i M ni W / to be an eigenvector of W n M M resp. We will
establish the following ErdsSchleinYau type delocalization theorem o analogous to . . before
reading the current one. Then with overwhelming probability. This result asserts assuming o
uniformly exponentially decaying distribution for the coecients that with overwhelming
probability. . all the unit eigenvectors of the Wigner matrix have coecients On/o thus the
energy of the eigenvector is spread out more or less uniformly amongst the n coecients. the
unit singular vectors ui . u . . which is an essential ingredient to Theorems . Observe from the
singular value decomposition that one can nd orthonormal bases u . Similarly. . b dened in
have all coecients uniformly of size OKn/ log n. We dene a right singular vector ui resp. vi are
determined up to multiplication by a complex phase ei . In the proof of the Four Moment
Theorem as well as the Gap Theorem for n n Wigner matrices in . Let gt be independent of n.
. The arguments in this paper are very similar to. a crucial ingredient was a variant of the
Delocalization Theorem of Erds. and let M obey condition C for some C . W n M M with
eigenvalue i . we shall primarily restrict attention to the generic case in which this vanishing
does not occur. such that M ui i vi and M vi i ui . Suppose further that that ij K almost surely
for some K gt which can depend on n and all i. vp Cp for the corange kerM of M and of Cp
respectively. and is also of some independent interest Theorem Delocalization theorem. .
When one just assumes uniformly bounded C moment rather than uniform exponential
decay. the bound becomes On/O/C instead where the implied constant in the exponent is
uniform in C . . TERENCE TAO AND VAN VU . . . . Proposition . . Schlein. all the unit left and
right singular vectors of M with eigenvalue i in the interval a. . we will need a Delocalization
theorem for the singular vectors of the matrix M . up Cn and v . lt lt . of course.
b is at least p.. The claim follows. unless otherwise stated. All statements here are
understood to hold only in the asymptotic regime when n is suciently large depending on all
quantities that are independent of n. Throughout this paper. is the number of eigenvalues in
I. We write X OY .g. We remark that a very similar result with slightly dierent hypotheses on
the parameters and on the underlying random variable distributions was recently established
in . y and C will remain other independent of n. or the matrix M will depend on n. Some
quantities e. and let gt be independent of n. We write X Y . Proposition Theorem Eigenvalue
concentration theorem..y x dx p. imposed so that the singular values are almost surely
simple. . As with other proofs of delocalization theorems in the literature. Let the hypotheses
be as in Theorem . while other quantities e. but anything which is polynomial in K and log n
will suce for our purposes.UNIVERSALITY FOR COVARIANCE MATRICES The factors K
log n can probably be improved slightly. or Y X if one has X CY for all suciently large n and
some C independent of n. then each event ij K with K n/C say occurs with probability On .
Note however that C is allowed to depend on other quantities independent of n. but in
practice we will be able to eliminate this hypothesis by a limiting argument as none of the
bounds will depend on any quantitative measure of this continuity. such as and y. Proof. we
will be able to apply the above theorem with K n/C without diculty. Let the hypotheses be as
in Theorem . We isolate one particular consequence of Theorem also established in
Corollary Concentration of the bulk. one has with overwhelming probability uniformly in I that
NI p where NI i p i W I I MP. Then there exists gt independent of n such that with
overwhelming probability. n will be an asymptotic parameter going to innity. b of length I K
log n/n. Then for any interval I a.g. Notation. From Theorem . Thus in practice. Observe that
if M obeys condition C. one has a i W b for all p i p. p. we see with overwhelming probability
that the number of eigenvalues in a . if is suciently small depending on . Theorem is in turn
deduced from the following eigenvalue concentration bound analogous to . We write X oY if
X cnY where cn as n . Y X. . X Y . Corollary . The continuity hypothesis is a technical one.
and A F trAA / for the Frobenius or HilbertSchmidt norm. Similarly. Let p n. then i Mn. thus
for instance if p/n y for some lt y then p n. We write X for the length of a vector X. TERENCE
TAO AND VAN VU or X Y if X Y X. and Mn. Lemma Weyl inequality. This follows from the
same minimax formulae used to establish Lemma . ii and iii follow from the minimax formula i
Mn. one can of course use the transpose of ii instead. If M.p is a p n matrix. For p n. Claim i
follows from the minimax formula i An V dimV i vV v inf sup v An v where V ranges over
idimensional subspaces in Cn .p for all i lt p.p i Mn. Tools from linear algebra. iii If p lt n.
Proof. in order to free up the letter i to denote an integer usually between and n.p v . A A op
for the operator norm of a matrix A. then i An i An i An for all i lt n. then i A i B A B op for all i
n. then i Mn. Proof. then i M i N M N op for all i p. N are p n matrices.. We write for the
complex imaginary unit.p for all i p.p is a p n minor. If A. i If An is an n n Hermitian matrix. In
this section we recall some basic identities and inequalities from linear algebra which will be
used in this paper. Basic tools . and An is an n n minor. with the understanding that Mn. B
are n n Hermitian matrices.p i Mn. We begin with the Cauchy interlacing law and the Weyl
inequalities Lemma Cauchy interlacing law.p is an pn minor.p . .p i Mn. . Let p n. ii If Mn.p i
Mn.p is a pn matrix. if Mn. and Mn.p V dimV inp vV v inf sup Mn.
where v Mp. .n i Mp. .n Y . un An Cn is an orthonormal eigenbasis corresponding to the
eigenvalues An . and let X u be a right unit singular vector of x Mp. .n .n Mp. Suppose that
none of the eigenvalues of An are equal to i An .n j j Mp. We omit the details.UNIVERSALITY
FOR COVARIANCE MATRICES Remark . and v y is a left unit singular vector of Mp. .n with
singular value i Mp. In a similar vein. Lemma . and let eigenvector of An with eigenvalue i An
. We have the following elementary formula for a component of an eigenvector of a Hermitian
matrix. .n X . where x C and u Cn . Then x n j j An i An uj An X . This implies an analogous
formula for singular vectors Corollary Formula for coordinate of a singular vector. .n are
equal to i Mp. One can also deduce the singular value versions of Lemmas .n are equal to i
Mp. . then y j Mp. See e.n . n . Suppose that none of the singular values of Mp.n . . vminp. if
Mp. and none of the singular values of Mp.n minp.n vj Mp. .n j j Mp. where x C and v Cn .n .n
. where y C and v Cp .n Y for some Y Cn .n with singular value i Mp. Let p.n i Mp.n Cp is an
orthonormal system of left singular vectors corresponding to the nontrivial singular values of
Mp. in terms of the eigenvalues and eigenvectors of a minor Lemma Formula for coordinate
of an eigenvector. . . n An of An . from their Hermitian counterparts by using the augmented
matrices .n uj Mp.n . . Then x j Mp.n Mp. .n minp. . Proof.g. where u An .n be a p n matrix for
some X Cp . and let Mp.n Mp. Let An An X X a v x be a unit be a n n Hermitian matrix for
some a R and X Cn .
The Stieltjes transform sz of a Hermitian matrix W is dened for complex z by the formula n .n
vj Mp.n Mp. We just prove the rst claim. one obtains the claim. .n Mp. sz n i i W z It has the
following alternate representation see e. we obtain x n j j Mp. Chapter Lemma . variance . n
Cn be a random vector whose entries are independent with mean zero. Proof. Observe that
is a unit eigenvector of the matrix x Mp. i Mp.n j Mp. Let X .n X Mp.. one has H X d OK log n .
so the claim follows.n Mp. .n uj Mp.n Mp. Tools from probability theory. TERENCE TAO AND
VAN VU where u Mp.n Mp. .n Mp. Proof. and vanishes for trivial ones. . . . Then t P H X d t
exp . .n Cn is an orthonormal system of right singular vectors corresponding to the nontrivial
singular values of Mp. Taking traces. kk za Wk zI ak k is the k th diagonal entry of W zI .
uminp.n Mp. and j Mp.jn be a Hermitian matrix.n . Let W ij i.n X .n Mp. where K E . . Applying
Lemma . as the second is proven analogously or by u taking adjoints. and let z be a complex
number not in the spectrum of W . and are bounded in magnitude by K almost surely for
some K.n for the minp.n Mp.n Mp. By Schurs complement.n .n . K In particular. Then we
have sn z n n k kk z a Wk zI ak k where Wk is the n n matrix with the k th row and column
removed. . .n . n nontrivial singular values possibly after relabeling the j. and ak Cn is the k th
column of W with the k th entry removed.n Mp. Let H be a subspace of dimension d and H
the orthogonal projection onto H.g.n j Mp. We will rely frequently on the following
concentration of measure result for projections of random vectors Lemma Distance between
a random vector and a subspace.n But uj Mp.n X X with eigenvalue i Mp.
But if i a .n n.n j n j Mp.n .n OK log n/n for all j J. b or that the nth coordinate x of ui is OKn/
log n holds with uniformly overwhelming probability. minp. We shall just establish the claim
for the right singular vectors ui . We can also assume that K log n on as the claim is trivial
otherwise. either i a . we conclude that j Mp. Delocalization The purpose of this section is to
establish Theorem and Theorem . j Mp. .n . .n OK log n.n by its transpose to return to the
regime p n. the proof is a short application of Talagrands inequality .UNIVERSALITY FOR
COVARIANCE MATRICES with overwhelming probability. .n K where M Mp. since i n i . .
one would have to replace M p. the lefthand side of is then bounded from below by n H X log
K n where H Cp is the span of the vj Mp.n i Mp. By Pythagoras theorem. We x and allow all
implied constants to depend on and y. The recent paper also contains arguments and results
closely related to those in this section. . Applying Corollary .. .n X . so that the singular
vectors ui are well dened up to unit phases. one can nd with uniformly overwhelming
probability a set J . or minp. it suces to show that with uniformly overwhelming probability. b .
n with J K log n such that j Mp. b . Fix i p. one has H X K log n In the case p n. vj Mp. as the
claim for the left singular vectors is similar.. Proof. Deduction of Theorem from Theorem . .
As M is continuous. Sections .. . Lemma . The material here is closely analogous to . as well
as that of the original results in . .n X log n j Mp.n for j J.n i Mp. and can be read
independently of the other sections of the paper. But from Lemma and the fact that X is
independent of Mp. In particular.n i Mp. it suces by the union bound and symmetry to show
that the event that i falls outside a . we see that the nontrivial singular values are almost
surely simple and positive. See . then by Theorem . We begin by showing how Theorem
follows from Theorem .
one concludes k C. which illustrates the techniques used to prove Theorem . and the claim
follows. we see that sz p p nMM which k . . of controlling the eigenvalue counting function NI
via the Stieltjes transform sz p p j . b of length I K log n/n. Let the hypotheses be as in
Theorem . Ima Wk zI ak k . and ak Cp is the k th column of W with the k th entry removed. j
W z NI p Fix I. To prove this proposition. A crude upper bound. kk z a Wk zI ak k where kk is
the kk entry of W .. we suppose for contradiction that NI CnI for some large constant C to be
chosen later. We follow the standard approach see e. this leads to a contradiction with
overwhelming probability. It thus remains to establish Theorem . and also plays an important
direct role in that proof Proposition Eigenvalue upper bound. We rst establish a crude upper
bound. We will show that for C large enough. and z x Imsz recall that p n so from one has
Imsz C. then for any interval I a . I/. Let the hypotheses be as in Theorem . TERENCE TAO
AND VAN VU with uniformly overwhelming probability. and NI was dened in . . one has with
overwhelming probability uniformly in I that NI nI where I denotes the length of I. Using the
crude bound Im z p p Imz and . with W replaced by the p p matrix W only has the nontrivial
eigenvalues. If x is the midpoint of I. we see that Applying Lemma .g. Wk is the p p matrix
with the k th row and column of W removed.
Imak Wk zI ak Mk Mk n where Xk Cn is the adjoint of the k th row of M . Fix k. We conclude
that j Mk Xk uj Mk . j Wk x p j a vj Mk k . . Xk uj Mk C jJ . one has p and thus a Wk k zI ak j p
a vj Mk k . p of length J Cn such that j Wk I. Thus. but this will not be of concern since all of
our claims will hold with overwhelming probability. . a vj Mk k Next. and Mk is the p n matrix
formed by removing that row. . . if we let v Mk . note that from and the Cauchy interlacing law
Lemma one can nd an interval J . n The advantage of this latter formulation is that the
random variables Xk and uj Mk are independent for xed k. up Mk Cn be coupled orthonormal
systems of left and right singular vectors of Mk . and let j Wk n j Mk for j p be the associated
eigenvectors. and thus n . vp Mk Cp and u Mk . . . . j Wk z Ima Wk k We conclude that zI ak j
a vj Mk k . there exists k p such that C. . Note that and ak Wk M k Xk n By the pigeonhole
principle. . . j Wk x C The expression a vj Mk can be rewritten much more favorably using as
k j Mk Xk uj Mk . n C jJ Since j Wk I.UNIVERSALITY FOR COVARIANCE MATRICES The
fact that k varies will cost us a factor of p in our probability estimates. . . one has j Mk n.
xz where we use the branch of y z yz with cut at a. where H is the span of the eigenvectors
uj Mk for j J.. We now begin the proof of Theorem in earnest. The strategy is to compare s
with the MarchenkoPastur Stieltjes transform sMP. for closely related lemmas.y x dx nI .
Then for any interval I in a .y y z yzsMP. Lemma . log .y z sz with uniformly overwhelming
probability for each z with Rez L and Imz . one has NI n with overwhelming probability. that
lemma was phrased for the semicircular distribution rather than the MarchenkoPastur
distribution. To put it another way. But from Lemma and we see that this quantity is n with
overwhelming probability.y z R MP. b that is asymptotic to y z as z . I MP. but an inspection
of the proof shows the proof can be modied without diculty. sMP. strictly speaking. Let / /n.
our arguments will not use any quantitative estimates on this continuity. Proof. We have the
following standard relation between convergence of Stieltjes transform and convergence of
the counting function Lemma Control of Stieltjes transform implies control on ESD. .y z gt .
We continue to allow all implied constants to depend on and y. This follows from . It suces by
a limiting argument using Lemma to establish the claim under the assumption that the
distribution of M is continuous. Reduction to a Stieltjes transform bound. Corollary .y z with
ImsMP. Suppose that one has the bound sMP. TERENCE TAO AND VAN VU The lefthand
side can be rewritten using Pythagoras theorem as H Xk . See also and . b with I max. This
concludes the proof of Proposition . for z in the upper halfplane. giving the desired
contradiction with overwhelming property even after taking the union bound in k.y z is the
unique solution to the equation sMP. and L.y z yz y z yz yz dx.y x A routine application of and
the Cauchy integral formula yields the explicit formula sMP. gt . Details of these
computations can also be found in . .
one has Yk y o y ozsz with overwhelming probability uniformly in k and I. We will shortly
show a similar bound for Yk itself Lemma Concentration of Yk . For this. and thus since the uj
are unit vectors p EYk Mk where sk z is the Stieltjes transform of Wk . . inserting the
identities . n j Wk z Suppose we condition Mk and thus Wk to be xed.y z o with uniformly
overwhelming probability. it suces to show that for each complex number z with a / Rez b /
and Imz K log n . For each k p. . we see that to show Theorem . Thus z p p p sk z sz O p p
and thus EYk Mk p p zsz O n n n y o y ozsz since p/n y o and / on. n one has sz sMP. we
return to the formula .UNIVERSALITY FOR COVARIANCE MATRICES In view of this
lemma. one has where Yk j sz p p k kk z Yk p j Mk Xk uj Mk . j j Mk n j Wk z p zsk z n p p j j
Wk z From the Cauchy interlacing law Lemma we see that the dierence p sk z sz p p j j W z j
j Wk z is bounded in magnitude by O p times the total variation of the function on . the entries
of Xk remain independent with mean zero and variance . which is O .
y z. cannot both hold for n large enough. we thus claim that either holds for all z in the
domain of interest. Thus the continuity argument shows that holds with uniformly
overwhelming probability for all z in the region of interest for n large enough. the two
solutions of are sMP. b. From one has sMP. one obtains kk sz p p k z y o y ozsz with
overwhelming probability. yz since y z yz has zeroes only when z a. By using a n net say yz
of possible zs and using the union bound and the fact that sz has a Lipschitz constant of at
most On say in the region of interest we may assume that the above trichotomy holds for all
z with a/ Rez b/ and Imz n say. one has either or or sz yz sMP. which implies that . which
gives and thus Theorem . or there exists a z such that as well as one of or both hold. from we
see that yz sMP. .y z from yz is bounded from yz below. cannot both hold for n large enough.
thus sz almost solves in some sense.y z yz y z yz .y z and yz yz sMP. sMP.y z yz yz which
implies that the separation between sMP. TERENCE TAO AND VAN VU Meanwhile.y z yz
sMP. and z is bounded away from these singularities. and so holds in this case.y z o with the
convention that yz when y .y z are both o. From the quadratic formula. One concludes that
with overwhelming probability. By continuity. Similarly. kk o with overwhelming probability
again uniformly in k and I. we see also that . When Imz n . then sz.y z o yz sz yz o yz sz
sMP. Inserting these bounds into . we have Xk n and hence by Lemma .
k.jn be matrix ensembles obeying condition C for some C . ij match to order . q .
.UNIVERSALITY FOR COVARIANCE MATRICES . . ij n/C almost surely for all i. . . . . if c is
suciently small depending on c . by truncation and adjusting the moments appropriately. our
bounds will not depend on any quantitative measure on the continuity. Thus. M is continuous.
. Next. . xk . . . Then for any i lt i lt ik n. and that ij and ij match to order . . xk R. q . . . . . . . We
begin by observing from Markovs inequality and the union bound that one has ij . . Theorem
Truncated Four Moment Theorem. . and such that G is supported on the region q . by a
further approximation argument we may assume that the distribution of M. For each i p. using
Lemma to absorb the error. . and so the general case then follows by a limiting argument
using Lemma . . whose proof is delayed to the next section. Let M ij ip. Qi M . . whose
eigenvalues are M . Qik M EG ni M . we introduce in analogy with the . ij n/C say for all i. nik
M . . . . and for n suciently large depending on . . . Let G Rk Rk R be a smooth function
obeying the derivative bounds j Gx . Proof of Theorem and Theorem We rst prove Theorem .
The key technical step is the following theorem. . The arguments follow those in . . For
suciently small c gt and suciently large C gt the following holds for every lt lt and k . . If ij . . .
Assume that p/n y for some lt y . Qik M nc . qk nc for all j and x . . . to ensure that the singular
values are almost surely simple. one may assume without loss of generality that ij . Qi M . .
This is a purely qualitative assumption. j. j with probability On . . and the gradient is in all k
variables. c we have EG ni M . . xk . . as well as . nik M . then the conclusion still holds as
long as one strengthens to j Gx . . . . . . .jn and M ij ip. . . Given a p n matrix M we form the
augmented matrix M dened in . . q . p M . qk nc . qk R. qk njc for some c gt . together with
the eigenvalue with multiplicity n p if p lt n. .
Clearly is convex. We set Qi M if the singular value i is repeated. The main observation here
is the following lemma. . . j M i M i M j j M i M p The factor of n in Qi M is present to align the
notation here with that in . qk . and compact support. in which one dilated the matrix by n.
variance one. Observe the upper bound Qi M n jpji np . we prove Theorem . To bound the
other term in . Accordingly. In order for to match to third order. but this event occurs with
probability zero since we are assuming M to be continuously distributed. Let be a complex
random variable with mean zero. and third moment bounded by some constant a. w ei z. . It
is also invariant under the symmetry z. ei w . Section . Then there exists a complex random
variable with support bounded by the ball of radius Oa centered at the origin and in
particular. one can now remove the hypothesis in Theorem that G is supported in the region
q . i M /n is bounded np away from zero. and that E E and E E . In particular. . and any p i p.
E where ranges over complex random variables with mean zero. it suces that have mean
zero. . Lemma . . Lemma Matching lemma. TERENCE TAO AND VAN VU arguments in the
quantities Qi M n i M i M n jpji np . obeying the exponential decay hypothesis uniformly in for
xed a which matches to third order. The gap property on M ensures an upper bound on Qi M
Lemma . By applyign a truncation argument exactly as in . variance . . and so ni M O/n. one
can now handle the case when G is independent of q . j M i M ni M From Corollary we see
that with overwhelming probability. If M satises the gap property.. then for any c gt
independent of n. unit variance. one has Qi M nc with high probability. Proof. Next. let C be
the set of pairs E . and Theorem follows after making the change of variables n and using the
chain rule and Corollary . qk nc . one repeats the proof of . . Proof. assuming both Theorems
and .
The proof of Theorem It remains to prove Theorem . . . and so again by convexity and
rotation invariance z . By Theorem .UNIVERSALITY FOR COVARIANCE MATRICES for
phase . by convexity cz. An inspection of the argument shows that if the third moment is
bounded by a then the support can also be bounded by Oa . w / whenever z cz and w w. . Qi
M . . ei/ w . . . and arbitrarily large third moment.e. nik M . EQi M Onc for some c gt
independent of n. for each i. it suces to establish a bound EG ni M . we can nd ij which
satises the exponential decay hypothesis and match ij to third order. By telescoping series.
The next two sections are devoted to the proofs of Theorem and Theorem . and hence by
convexity and rotation invariance . w . Since z. . as is the case here. Qi M . whenever z cz. .
Now consider a random matrix M as in Theorem with atom variables ij . variance one..
Remark . we omit the details. It is easy to construct complex random variables with mean
zero. By the above lemma. so by Theorem one has EQi M Onc for some c gt independent of
n. Qik M nc . Since the third moment is comparable to z w. i. every complex random variable
with nite third moment with mean zero and unit variance can be matched to third order by a
variable of compact support. ei/ w . only a uniformly bounded C moment for C large enough
is required. and let p i p. nik M . . . . Because of this. respectively. . w and . The above trick to
remove the exponential decay hypothesis for Theorem also works to remove the same
hypothesis in . lies in it also for some absolute constant c gt . compact support. . one can
replace all the exponential decay hypotheses in the results of . Let q be a smooth cuto to the
region q nc for some c gt independent of n. w whenever w w. Theorem . Section . by a
hypothesis of bounded C moment. we thus conclude that contains all of C . and hence by
convexity any . Qik M EG ni M . . . w both lie in . Thus. By Lemma . The point is that in the
analogue of Theorem in that paper implicit in . if z. . then z. One last application of convexity
then gives z /. M satises the gap property. the exponential decay hypothesis is not used
anywhere in the argument. We conclude that M also obeys the gap property. . j.
Let the notation and assumptions be as in Theorem . There exists a positive constant C
independent of k such that the following holds. whenever Pij . . Proposition is trivially true if is
comparable to n. TERENCE TAO AND VAN VU under the assumption that the coecients ij .
resp. where Mz is the augmented matrix of M z. . say the qr entry for some q p and r n.
except for the qr entry. Pij . if C is suciently large depending on cf. We condition i.e. . and that
the eigenvalues are given up to unit phases by . . uj M z Note also that the analogue of in .
eq . In view of the above proposition. e uij M z n/ . er are a good conguration for i . vij M z Cp
are unit right and left singular vectors of M z. Let gt . we see that to conclude the proof of
Theorem and thus Theorem it suces to show that for any gt . freeze all the entries of M z to
be constant. . M zer / n/ . Proof. dened in . This follows by applying . . Proposition to the p n p
n Hermitian matrix Az nMz. . ij of M and M are identical except in one entry. . We apply the
following proposition. we have ni M z nij M z n i ij . . Write M z for the matrix M or M with the
qr entry replaced by z. then e vij M z. so one can restrict attention to the regime on.
Proposition . . Pij . we assume Delocalization at ij If uij M z Cn . r q For every Pij . . which
follows from a lengthy argument in Proposition Replacement given a good conguration. . np
M z vj M z and . which is z. that M . er are a good conguration for i . . nij Az n n. then we have
if ij and ij match to order . ik with overwhelming probability. left singular vectors vi M z
corresponding to singular values i Az with i ij lt . we have Singular value separation For any i
n with i ij n . eq . ik if the above proper ties hold. or if they match to order and holds. Note that
the eigenvalues of Az are n M z. . Also. Assuming this good conguration. We say that M . M
zeq . We assume that for every j k and every z n/ whose real and imaginary parts are
multiples of nC . is the orthogonal projection to the span of right singular vectors ui M z resp.
since the claim then follows by interchanging each of the pn On entries of M into M
separately.
Corollary and is omitted. This gives the claim. Suppose that p / p lt p and l p/ is such that gi
.n N i Mp. It will suce to show that The main tool for this is Lemma Backwards propagation of
gap. as does . Using the exponential decay condition. and write N p n. and also x the z n/
whose real and imaginary parts are multiples of nC .. Corollary also gives with overwhelming
probability. By a limiting argument we may assume that M has a continuous distribution.p .n
backwards in the p variable. we need a slight variant Proposition . we let Mp.n . A similar
argument using Pythagoras theorem and Corollary gives with overwhelming probability
noting as before that we may restrict attention to the regime on.l. and we dene following the
regularized gap N i Mp. The conclusions of Theorem and Theorem continue to hold if one
replaces the qr entry of M by a deterministic number z On/O/C . closely following the
analogous arguments in . By the union bound again and Proposition . We return to the task
of establishing a good conguration with overwhelming probability.UNIVERSALITY FOR
COVARIANCE MATRICES Our main tools for this are Theorem and Theorem . we may
truncate the ij and renormalise moments. the eigenvalue separation condition holds with
overwhelming probability for every i n with i j n if C is suciently large. gi. for any i l lt i p p . so
that the singular values are almost surely simple. p instead of p. By the union bound we may
x j k. where C gt is a large constant to be chosen later.p nc .p inf . . Actually. until one can
use Theorem to show that the gap occurs with small probability. the strategy is to propagate
a narrow gap for M Mp . using Lemma to assume that ij logO n almost surely. logC N log gi .
N i illtii p mini i .n be the p n matrix formed using the rst p rows of Mp . As in . Proof of
Theorem We now prove Theorem .l. More precisely. This is proven exactly as in . We write i
instead of i. and Theorem follows.n .
/ v Large row We have n exp log. . rather than equal.n m/ n log. p Mp. n p are only
comparable up to constants. ii Small inner products There exists p/ i i l lt i i /p with i i logC / n
such that i jlti Xp uj Mp.n .n be an orthonormal system of right singular vectors of Mp.n /
explog. / vi Large inner product near i There exists p/ i /p with i i logC n such that Xp Xp ui
Mp.p m / . .n Strictly speaking. but these adjustments make only a negligible change to the
proof of that lemma. there are some harmless adjustments by constant factors that need to
be made to this lemma. Proof. n .n .l. ultimately coming from the fact that n.n Mp. .n . ni i . up
Mp.n which after removing the bottom row and rightmost column which is Xp .n . n.l. . .
TERENCE TAO AND VAN VU for some lt which can depend on n.n i i . p.n . n iii Large
singular value For some i p one has n exp log. Mp. This follows by applying . n Xp ui Mp. . .n
Apn n . i Mp. and that for some m with gi .n associated to Mp.n ni Mp. . . and let u Mp. plus p
zeroes yields the p n p n Hermitian matrix Apn n Mp. Let Xp be the pth row of Mp . m/ log.p
m gi . n . Then one of the following statements hold i Macroscopic spectral concentration
There exists i lt i p with i i logC / n such that ni Mp.n / iv Large inner product in bulk There
exists p/ i /p such that exp log. Lemma to the p n p n Hermitian matrix Mp.
. v in Lemma all fail with high probability. B.n . . New York University. But Proposition can be
proven by repeating the proof of . Linear Algebra Appl. Universality for mathematical and
physical systems.n and . The large vj Mp.n is conditioned as in b. iii. Press. Zeitouni. . Beijing
. Appl. c Furthermore. B. to appear in Annals of Probability. . Costello and V. Yau. . American
Mathematical Society. Matrix Anal. . Guionnet and O. Orthogonal polynomials and random
matrices a RiemannHilbert approach. then ii and vi in fact occur with a conditional probability
of at most m logC n n . Math. almost verbatim. . Eigenvalues and condition numbers of
random matrices. Schlein and HT. and an orthonoruj Mp. International Congress of
Mathematicians Vol. no. . A. it then suces to show that Proposition Bad events are rare. K. to
appear in Comm. Zrich.n for p/ j /p are of magnitude at most n/ logC n with overwhelming
probability. Pure Appl. L.n coecient event in . Science Press. C such that if l C and Mp. Then
a The events i. Erds. . Courant Institute of Mathematical Sciences. References G. there is a
constant C depending on C . Math. Edelman. P. P. Mathematics Monograph Series . SIAM J.
Comm. Vu. L. Proposition respectively. and set n for some suciently small xed gt . An
introduction to random matrices.n to be a matrix with this property. to appear in SIAM
discrete math. Section . RI. Schlein and HT. New York. Deift. . Erds. I. G. Concentration of
random determinants and permanent estimators.UNIVERSALITY FOR COVARIANCE
MATRICES which has eigenvalues n Mp. Bai and J. Conditioning Mp. Pch. iv. Suppose that
p / p lt p and l p/. Local semicircle law and complete delocalization for o Wigner random
matrices. b There is a constant C such that all the coecients of the right singular vectors uj
Mp. .n mal eigenbasis that includes the vectors for j p. np Mp. book to be published by
Cambridge Univ. Providence. . the events ii and vi occur with a conditional probability of at
most m n . Silverstein. . Theorem and . Courant Lecture Notes in Mathematics. Math. Soc. A.
Ben Arous and S. the only signicant dierence being that Theorem and Theorem are applied
instead of . D. as Apn has zero diagonal. By repeating the arguments in . Yau. Proposition
with only cosmetic changes. A. . Phys. Semicircle law on short scales and delocalization of o
eigenvectors for Wigner random matrices. Anderson. The distribution and moments of the
smallest eigenvalue of a random matrix of Wishart type. Spectral analysis of large
dimensional random matrices. Lemma iii cannot occur here. Eur. . Universality of local
eigenvalue statistics for some sample covarie e ance matrices. . . Z. Edelman.. Deift.
Electron. . Phys. . Soc. Pch. Nuclear Phys. . The distribution of the rst eigenvalue spacing at
the hard edge of the Laguerre unitary ensemble. . Wegner estimate and level repulsion for
Wigner random o matrices. Statistical Ensembles of Complex. Ramirez. Kyushu J. Pastur. to
appear in Comm. Guionnet and O. N. Schlein and HT. . V. The distribution of eigenvalues in
certain sets of random matrices. Statist. Comm. J. preprint. M. N. A universality result for the
smallest eigenvalues of certain sample covariance matrices. . Nagao. Inst. P. Mehta and M.
AMS . Feldheim and S. Correlation functions of random matrix ensembles related to classical
orthogonal polynomials. Kluwer Academic Publishers. Japan . . K. Erds. S. A. Gustavsson.
Random matrices. A. Universality of the local spacing distribution in certain ensembles of
Hermitian Wigner matrices. Eine neue Herleitung des Exponentialgesetzes in der
Wahrscheinlichkeitsrechnung. Lett. Math. B . J. Academic Press. Ramirez. arXiv. Yau and J.
Yin. New York. Katz and P. Rackauskas. Gaudin. TERENCE TAO AND VAN VU L. Math. L.
V. Erds. . no. arXiv. A. B. Universality of sinekernel for Wigner matrices o with a small
Gaussian perturbation. Ginibre. . . M. . Gtze. Erds. Math. M. Mehta. volume . Yau. Comm.
HT. Erds. Johnstone. P. . . L. Nuclear Phys. B. Forrester. Slevin. I. Forrester. Probab. Rider.
Yau. American Mathematical Society.L. Rate of convergence in probability to the
MarchenkoPastur law. H. P. no. no. F. Surveys. . . Erds. Universality at the edge of the
spectrum in Wigner random matrices. J. Forrester. Journal of Mathematical Physics .
Marchenko. Bulk universality for Wigner matrices. no. T. The spectrum edge of random
matrix ensembles. O. no. . . no. Gaussian uctuations of eigenvalues in the GUE. Bulk
universality for Wigner o hermitian matrices with subexponential decay. L. . and HT. Sarnak.
no. electronic. RI. A. and Real Matrices. . Statist. Frobenius eigenvalues. V. o arXiv. SFB
Universitt o a Bielefeld. Providence. Preprint . J. T. . . Tikhomirov. . K. B. Math. Tikhomirov.
Random Matrices and the Statistical Theory of Energy Levels. Phys. Poincar Probab.
Quaternion. Soshnikov. . On the distribution of largest principal component. . J. Math. no. .
Schlein and HT. J. Ann. Laguerre ensembles of random matrices nonuniversal correlation
functions. Vu. The local relaxation ow approach to universality o of the local statistics for
random matrices. Approximation theory in the central limit theorem. On the density of
eigenvalues of a random matrix. . Russian Math.L. Z. F. The concentration of measure
phenomenon. Phys. to appear in Math. J. . Ann. J. J. Phys. J. Spectra of random selfadjoint
operators. Math. S. N. Pastur. Paulauskas and A. Ramirez. . . . Witte. A. A. T.. Johansson. .
Rate of convergence to the semicircular law. B. Ramirez. Exact results and universal
asymptotics in the Laguerre random matrix ensemble. Yau. J. submitted. NY. B. Sodin.
Schlein and HT. Ledoux. Math. Tao. . . . Wadati. Gtze. . Diusion at the random matrix hard
edge. Nagao. L. Mat. . B. Sb. . .. . Schlein. . American Mathematical Society Colloquium
Publications. W. Lindeberg. Universality in the bulk of the spectrum for complex sample
covariance matrices. Schlein. o Bernoulli . e e preprint. . Comm. A. M. L. and monodromy.
Phys. J. J. Zeitouni. Yau. Res. Concentration of the spectral measure for large matrices.
Phys. Mathematical survey and monographs. L.
. .rutgers. The Annals of Mathematics . Strong limits of random matrix spectra for sample
covariance matrices of independent elements. W. . . Phys. no. . M. Wachter. Rutgers. On the
distribution of the roots of certain symmetric matrices. Piscataway. Goldstine. to appear in
Annals of Probability. Math. A note on universality of the distribution of the largest
eigenvalues in certain sample covariance matrices. Y. no. Limiting spectral distribution for a
class of random matrices. Vu. Soshnikov. no. K. Vu. Phys. . T. Soc. Tao and V. On random
matrices singularity and determinant. Random matrices Universality of ESDs and the circular
law. Soshnikov. .. Tao and V. .. Vu. Random Structures Algorithms . . Multivariate Anal.
Bull.UNIVERSALITY FOR COVARIANCE MATRICES A. T. UCLA. . Los Angeles CA Email
address taomath. Wigner. Numerical inverting matrices of high order. Probab.edu
Department of Mathematics. Random matrices The distribution of the smallest singular
values Univerality at the Hard Edge. Probab. Math. Tao and V. Vu with an appendix by M. J.
NJ Email address vanvumath. . . . von Neumann and H. to appear in Comm. Random
matrices universality of local eigenvalue statistics up to the edge. Random matrices The local
statistics of eigenvalues. Gaussian limit for determinantal random point elds. To appear in
GAFA. Vu. Tao and V. Ann. . J. J. A new look at independence. T. . Q.. Ann.edu . T. A.ucla.
Krishnapur. Department of Mathematics. . P. . to appear in Acta Math. Yin. Amer. Prob.
Statist. Ann. Tao and V. T. Talagrand.