Mathematical Physics, Analysis and Geometry - Volume 3

394 Pages·2000·2.62 MB·English

by  A. Boutet de Monvel, H. McKean (Editors)

Mathematical Physics, Analysis and Geometry 3: 1–31, 2000.
1
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Asymptotic Distribution of Eigenvalues of Weakly
Dilute Wishart Matrices
1 2
A. KHORUNZHY and G. J. RODGERS
1Institute for Low Temperature Physics, Kharkov 310164, Ukraine.
e-mail:
2 A. KHORUNZHY AND G. J. RODGERS
1.1. STRONG DILUTION AND SEMICIRCLE LAW
In papers [17, 19] we have studied the limiting eigenvalue distribution of large ran-
dom matrices that are strongly diluted. These are determined as the N-dimensional
matrices that have, on average, pN nonzero elements per row and 1 � pN � N
as N ! 1. We proved that under natural conditions the limiting eigenvalue dis-
tribution of strongly dilute random matrices exists and coincides with the Wigner’s
famous semicircle law [32]. The semicircle law is also valid when the entries of
the dilute random matrix are statistically dependent random variables [16, 20].
This case is of special interest in applications (see, for instance, [1, 7, 10, 13]).
In the pure (nondilute) regime these matrices have singularities in the eigenvalue
distribution. The strong dilution removes this singularity because the density of the
semicircle distribution is bounded. It should be noted that the Wigner’s semicircle
distribution is typical for large random matrices with jointly independent entries.
Therefore we have conjectured that the semicircle law arises in the ensembles of
[16, 20] because the strong dilution eliminates the statistical dependence between
random matrix entries. The same reasoning can explain the disappearance of the
singularity in the eigenvalue distribution. However, the last conjecture is not true.
1.2. WEAKLY DILUTE WISHART MATRICES
To study the transition to the semicircle law under dilution, we pass to the case
of weakly dilute random matrices. This means that we are now interested in the
asymptotic regime when pN D qN, q > 0 as N ! 1. In this case the statistical
dependence between random matrix entries persists, provided it exists in the pure
(nondilute) ensemble. We consider two random matrix ensembles with different
types of statistical dependence between the entries. These are the Wishart random
matrices HN and Gaussian random matrices AN with correlated entries. The first
ensemble is widely known in multivariate statistical analysis (see, e.q., [1]). Recent
applications of these matrices are related with the the theory of disordered spin
systems of statistical mechanics [14, 26, 24] and learning algorithms of memory
models of neural network theory [2, 13, 14]. The entries fHN.x; y/g are statistically
dependent (but uncorrelated) random variables. The degree of the dependence be-
tween HN.x; y/ and HN.s; t/ does not relate to the ‘distance’ jx � sj C jy � tj. In
contrast, correlations between matrix elements AN.x; y/ and AN.s; t/ in the sec-
ond ensemble we consider decay when the distance between them increases. Due
to this property, fANg can be regarded as the ensemble intermediate betweem ran-
dom matrix models with strongly correlated entries (see, e.g., [7, 10]) and random
matrices with independent entries.
1.3. MAIN RESULTS AND STRUCTURE OF ARTICLE
We study the limit of the normalized eigenvalue counting function of weakly dilute
real symmetric matrices HN and AN as N ! 1. We derive explicit equations for
EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES 3
the Stieltjes transform of the limiting distribution functions and determine recurrent
equalities for their moments. Basing on these relations, we study the properties of
the limiting eigenvalue distributions.
We show that both these distributions are different from the semicircle law. We
prove that, nevertheless, the singularity disappears from the limiting eigenvalue
density and that this happens for arbitrary values of q < 1.
Thus, our principal conclusion is that the singularity of the eigenvalue distrib-
ution is rather unstable under dilution and is destroyed even when this dilution is
weak.
To complete this introductory section, let us note that our results can be regarded
as generalizations of the statements proved for strongly dilute random matrices in
[19] and [20]. In this paper we use the technique developed in [20]. However, the
case of weak dilution studied here is more complicated and requires more accurate
analysis.
The paper is organised as follows. In Section 2 we present our main results
for the Gaussian random matrices AN with correlated entries. In Section 3 we
consider the weak dilution of the Wishart matrices HN. We prove the existence of
.i/
the limiting eigenvalue distributions � , i D 1; 2 in the limit N ! 1 (Theorems
2.1 and 3.1). At the second part of each of these sections we formulate Theorems
.i/
2.2 and 3.2 concerning the properties of the respective � . In Sections 4 and 5
we derive the main equations that determine the eigenvalue distributions of AN
and HN, respectively, as N ! 1. Section 6 contains the proofs of Theorems 2.2
.i/
and 3.2. We also present there the recurrent relations for the moments of � and
.i/
make conclusions about the support of the measure d� . In Section 7 we give a
summary of our results.
2. Gaussian Random Matrices with Correlated Entries
Let us consider N � N symmetric random matrices
1
AN.x; y/ D p a.x; y/; x; y D 1; : : : ; N; (2.1)
N
where random variables fa.x; y/; x 6 y; x; y 2 Ng have a joint Gaussian distrib-
ution. We assume that fa.x; y/g satisfy the following conditions:
Ea.x; y/ D 0; (2.2a)
Ea.x; y/a.s; t/ D V .x � s/V .y � t/ C V .x � t/V .y � s/; (2.2b)
where the sign E represents the mathematical expectation with respect to the mea-
sure generated by the a.x; y/’s and V .x/ is a nonrandom function such that V .�x/
D V .x/ and V is nonnegatively defined. Then the right-hand side satisfies condi-
tions for the covariance of random variables (see Lemma 4.5 at the end of Sec-
4 A. KHORUNZHY AND G. J. RODGERS
tion 4). The eigenvalue distribution of the random matrix ensemble (2.1), (2.2),
where V .x/ satisfies the condition
X
jV .x/j � Vm < 1; (2.3)
x
was studied in [18]. This case is known as weakly correlated random variables.
Indeed, condition (2.3) implies decay of the correlations between random matrix
entries AN.x; y/ and AN.s; t/ that are situated far enough from each other in the
matrix.
2.1. ENSEMBLE AND MAIN EQUATIONS
In this section we consider the ensemble of real symmetric random matrices
.q/ 1
A
N D p a.x; y/�xy; x; y D 1; : : : ; N; (2.4)
N
where a.x; y/ are the same as in (2.1) and the random variables f�xy; x 6 yg are
both independent between themselves and independent from fa.x; y/g. We assume
that �yx D �xy and the random variables have the common probability distribution
�
1 1; with probability q,
�xy D p (2.5)
q 0; with probability 1 � q.
Our main result concerns the normalized eigenvalue counting function (NCF) of
.q/
A given by the formula
N
� � � �
.q/ .N/ �1
� �IA D # � 6 � N ; (2.6)
N j
.N/ .N/ .q/
where � 6 � � � 6 � are eigenvalues of A .
1 N N
THEOREM 2.1. Assume that (2.3) holds. Then
.q/
(i) given q 2 .0; 1/ the NCF �.�IA / weakly converges in probability as N !
N
1 to a nonrandom function �q.�/;
R
�1
(ii) the Stieltjes transform fq.z/ D .� � z/ d�q.�/, Imz D6 0 can be found
from the the system of equations
Z
1
fq.z/ D gQq.pI z/ dp; (2.7a)
0
� Z
1 ��1
2
Q Q
gQq.pI z/ D � z � .1 � q/v fq.z/ � qV .p/ V .r/gQq.rI z/ dr ; (2.7b)
0
where
X
Q
V .p/ D V .x/Ef2�ipxg;
x2Z
EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES 5
and
Z
1
Q
v � V .0/ D V .p/ dpI
0
(iii) system (2.7) is uniquely solvable in the class 0 of functions g.pI z/, p 2
.0; 1/; z 2 C� analytical in this region and such that
Im g.pI z/Im z > 0; z 2 C�: (2.8)
Remarks. (1) Here and below we mean by the weak convergence of nonnega-
tive nondecreasing functions � .�IAN/ the weak convergence of the corresponding
measures
Z Z
1 1
1
lim ’.�/ d� .�IAN/ D ’.�/ d� .�/; ’ 2 C
0 .R/:
N!1
�1 �1
Generally the convergence of integrals can be regarded as convergence in average,
in probability or with probability 1.
(2) Each function gO 2 0 determines a nonnegative nondecreasing function �O .�/
R
�1
such that [11] gO.z/ D .� � z/ d�O .�/ and
Z
b
1
�O .a/ � �O .b/ D lim ImgO.� C i�/ d�: (2.9)
�#0 �
a
Relation (2.9), known as the inversion formula for the Stieltjes transform, is valid
for all a; b such that �O is continuous at these points.
Theorem 2.1 is proved in Section 4. Basing on (2.7), one can study the proper-
ties of �q.�/.
2.2. LIMITING EIGENVALUE DISTRIBUTION
To discuss the consequences of Theorem 2.1, let us note first that relations (2.7)
considered with q D 0 can be reduced to the equation
1
f0.z/ D : (2.10)
2
�z � v f0.z/
This equation is uniquely solvable and determines the Wigner semicircle distribu-
tion �0.�/ [32] with the density
�p
1 2 2
0 4v � � ; if j�j 6 2v,
%0.�/ � �
0.�/ D (2.11)
2�v 0; otherwise.
This observation shows that Theorem 2.1 generalizes the results of paper [20],
where the eigenvalue distribution of the ensemble (2.4) has been studied under con-
.N/ 1=2 �1=2
dition that random variables �xy are replaced by �O
xy that take values N p
with probability p=N and 0 with probability 1 � p=N. Then the limit p;N ! 1
6 A. KHORUNZHY AND G. J. RODGERS
considered in [20] corresponds to subsequent limiting transitions N ! 1 and
q ! 0. Another limiting transition q ! 1 in (2.7) leads to equations
Z
1
f1.z/ D g1.pI z/ dp;
0
� Z �
�1
Q Q
g1.pI z/ D � z � V .p/ V .r/g1.rI z/ dr :
This system has been derived in [18] the the limit f1.z/ of the Stieltjes transforms
of the NCF �.�IAN/ of the ensemble determined by (2.1), (2.2), and (2.3) (see
also [3, 4] for others and more general ensembles).
It should be noted that if
Z
1
dp
D 1; (2.12)
Q
0 V .p/
then corresponding to f1.z/, measure �1.d�/ has an atom at the origin. Indeed, one
can easily derive from (2.7) and (2.9) that if (2.12) holds, then
lim Imf1.i"/ D 1
"#0
(see also [4]).
Relation (2.12) is known in stochastic analysis as the interpolation condition for
the infinite sequence of Gaussian random variables f� .x/; x 2 Zg with zero aver-
age E� .x/ D 0 and covariance E� .x/� .y/ D V .x � y/ [22]. If (2.12) holds, then
the sequence f� .x/g can be regarded as insufficiently random. Apparently, random
dilution makes the sequence of such random variables ‘more disordered’. This
observation can be regarded as a heuristic explanation of the following proposition.
0
THEOREM 2.2. If q < 1, then the density %q.�/ D �
q.�/ exists and is bounded
p
everywhere by 1=.�v 1 � q/.
Theorem 2.2 is proved in Section 6. There we also derive and analyse recurrent
R
k
relations for the moments Lk D � d�q.�/. Basing on these relations, we study
the support of the measure d�q.�/.
3. Weak Dilution of Wishart Matrices
In this section we study the eigenvalue distribution of symmetric matrices
N
X
.q/ 1
H
N .x; y/ D ��.x/��.y/�xy; x; y D 1; : : : ; N; (3.1)
N
�D1
where f��.x/; x; � 2 Ng are independent random variables having joint Gaussian
distribution. We assume that these random variables satisfy conditions
E��.x/ D 0; E��.x/��.y/ D �xy���; (3.2)
EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES 7
where �xy is the Kronecker �-symbol;
�
1; if x D y,
�xy D
0; if x D6 y.
�
We assume also that �xy D �yx and f�xy; x 6 y are i.i.d. random variables (in-
dependent of fH�g) that have probability distribution (2.5). Thus, (3.1) represents
the weak dilution of random matrices
m
X
1
HN;m.x; y/ D ��.x/��.y/; x; y D 1; : : : ; N (3.3)
N
�D1
known since 30s in the multivariate statistical analysis as the Wishart matrices [1].
Being at present of considerable importance in this field, the ensemble (3.3) is
extensively studied in the statistical mechanics of the disordered spin systems (see,
e.g., [6, 26, 31] for rigorous results). Another important application of (3.3) is
related with the neural network theory, where HN are used as the interation ma-
trix of learning algorithms modelling auto-associative memory. In this approach,
N-dimensional vectors �� are regarded as the patterns to be memorised by the
system [13]. Dilution versions of (3.3) are important in this field of applications as
the models that can be tuned to give more precise correspondence with real systems
(see, e.g., [2]). These models are mostly studied in the regime of strong dilution
[5, 30]. The following statement concerns the normalized eigenvalue counting
function (2.6) of of weakly dilute random matrices (3.1), (3.2).
.q/
THEOREM 3.1. For each fixed q 2 .0; 1/ the NCF �.�IH / converges in the
N
limit N;m ! 1;m=N ! c > 0 to a nonrandom function �q;c. The Stieltjes
transform fq;c.z/ of �q;c.�/ satisfies equation
�
2p ��1
cu q
4
fq;c.z/ D � z � cu .1 � q/fq;c C p : (3.4)
2
1 C u qfq;c.z/
This equation is uniquely solvable in the class of functions 0 determined in Theo-
rem 2.1 and satisfying (2.8).
We prove Theorem 3.1 in Section 5. Regarding (3.4), one can easily observe
that, in complete analogy with (2.7), this equation determines a ‘mixture’ of two
equations: the one for the semicircle distribution with q D 0 (cf. with (2.10))
1
f0;c.z/ D
4
�z � cu f0;c.z/
and the equation (q D 1)
� �
2 �1
cu
f1;c.z/ D � z C (3.5)
2
1 C u f1;c.z/
8 A. KHORUNZHY AND G. J. RODGERS
derived in [23] for the Stieltjes transform of � .�IHN;m/ in the limit N;m ! 1,
m=N ! c > 0.
Corresponding to (3.5) eigenvalue distribution has the density given by the
formula [23]
q
d�1;c.�/ 1 � �2
2 2
D T1 � cUC�.�/ C 4cu � � � .c C 1/u ; (3.6)
2
d� 2�u �
where TxUC D max.0; x/ and �.x/ is the Dirac delta function. Let us stress that
if c < 1, then the density of �1;c.�/ has the singular component at the origin.
The following statement shows that this singularity disappears in the weak dilution
regime.
THEOREM 3.2. If q < 1 then the density of �q;c.�/ determined by fq;c.z/ (3.4)
p
2
is bounded from above by 1=u c.1 � q/ for all c > 0.
We prove Theorem 3.2 in Section 6. In this section we also derive recurrent
R
k
relations for the moments Lk D � d�q;c.�/.
4. Proof of Theorem 2.1
In this section we use the resolvent method developed in a series of papers (see,
e.g., [18]) and improved in [15]. This method is based on the derivation of relations
for the moments of the normalized trace of the resolvent of random matrix AN
X
1 1
�1
fN.z/ � TrGN.z/ D GN.x; xI z/; GN.z/ D .AN � z/ : (4.1)
N N
xD1
The important and in certain sense characteristic property of random matrices is
that EfN.z/ converges as N ! 1 to the variable f .z/ and one can derive closed
equations for it. The variance of fN.z/ vanishes as N ! 1 that means that the
normalized trace (4.1) is the self-averaging random variable.
The case of weakly diluted random matrices AN;q (2.7) is more complicated.
The average EGN.x; y/ is expressed in terms of the generalized ‘trace’
N
X
1
TN.x; y/ D �xrG.r; s/Vrs�sy; (4.2)
N
r;sD1
where we denoted
.q/ �1
G.x; y/ � G
N .z/ D .AN;q � z/
and Vxy � V .x � y/. Relations for the limit of EGN.x; y/ involve the limit of the
average ETN.x; y/.
Our main observation is that this pseudo-trace TN.x; y/ is also the selfaveraging
variable as N !1. We obtain expression for t .x; y/ D limN!1 ETN.x; y/.
EIGENVALUES OF WEAKLY DILUTE WISHART MATRICES 9
In contrast with the strong dilution, matrix t .x; y/ is nonzero and its entries take
two different values depending on whether x D y or not. These two limits t1 and
t2 involve explicitly parameter q and t1 � t2 vanishes for q ! 1. This determines
the difference between equations that we obtain for weakly dilute random matrices
and those derived for the corresponding pure (nondilute) ensemble.
To make the derivation self-consistent, we recall the basic elements of the method
developed in [15, 18].
O
Given two symmetric (or Hermitian) operators H and H acting in the same
�1 O O �1
space, the resolvent identity holds for G D .H � z/ and G D .H � z/
O O O
G � G D �G.H �H/G: (4.3)
This identity leads to two important observations; that the dependence of the re-
solvent on the random matrix can found explicitly and that this dependence is
expressed in terms of the resolvent (see formulas (4.7) and (4.9)). These two prop-
erties make the resolvent approach a fairly powerful tool in the spectral theory of
random matrices.
We prove Theorem 2.1 by showing the convergence of the trace fN;q.z/ D
�1 .q/
N TrG .z/;
N
lim EhfN;q.z/i D f
q;VQ .z/; (4.4)
N!1
and
� �
2
lim E jfN;q.z/ � EhfN;q.z/ij (4.5)
N!1
for all
� �
�1
z 2 3q D z 2 C; jImzj > 2Vmq C 1 ; (4.6)
where we denoted by h�i the mathematical expectation with respect to the measure
generated by random variables f�xyg.
On this way, we derive equation for f
q;VQ .z/ that determines the limiting eigen-
value distribution. Taking into account that the normalized trace of the resolvent
(4.1) is the Stieltjes transform of the NCF, we conclude that relations (4.4) and (4.5)
imply the weak convergence in probability of � .�IAN;q/, in the limit N ! 1, to
Q
a nonrandom function �q.�I V /. This can be proved by the usual arguments of the
theory of Herglotz functions. The reasonong is based on weak compactness of the
family � .�IAN;q/ and the Helly theorem [11] (see, e.g., [18] for more details).
Thus, relations (4.4) and (4.5) prove items (i) and (ii) of Theorem 2.1. Item (iii)
is proved in Lemma 4.4 (see the end of this section).
We split the remaining part of this section into three subsections. In the first
one we derive main relations that lead to the equation for f
q;VQ .z/. In the second
subsection we derive relations leading to the proof of (4.5). The third subsection
contains the proofs of the auxiliary facts and estimates.
10 A. KHORUNZHY AND G. J. RODGERS
4.1. DERIVATION OF MAIN RELATIONS
O
Let us consider identity (4.3) with H D AN;q and H D 0;
� �
X
�1=2
EhG.x; y/i D ��xy � �N EG.x; s/a.s; y/�.s; y/ ; (4.7)
s
�1
where � D �z . In (4.7) and below, we omit the varable z and subscripts q;N
and do not indicate limits of summations, if no confusion can arise.
To compute the average EG.x; s/a.s; y/, we use the following elementary facts
(see also [18]). It is related to the Gaussian random vector � D .�1; : : : ; �k/ with
zero average:
k � �
X
@F
E�jF.�1; : : : ; �k/ D E�j�lE ; (4.8)
@�l
lD1
where F is a nonrandom function such that all integrals in (4.8) exist. This formula
can be proved by using the integration by parts technique.
We will also use the formula that is a direct consequence of identity (4.3);
@G.x; s/ 1
D �p G.x; p/G.r; s/�pr : (4.9)
@a.p; r/ N
Now we can write that
1 X� �
EG.x; s/a.s; y/ D �p VpsVry C VpyVrs EG.x; p/G.r; s/�pr :
N
x;p
Substituting this relation into (4.7), we obtain equality
X � �
.1/
EhG.x; y/i D ��xy C � EhG.x; p/T .p; y/iVpy C �E
N .x; y/ ; (4.10)
p
where T is determined in (4.2) and we denoted
X
1
.1/

N .x; y/ D G.x; p/�prG.r; s/Vps�syVry: (4.11)
N
p;r;s
.1/
In the last part of this section we prove that vanishes in the limit N !1 (see
N
Lemma 4.1).
Turning back to (4.7), we can write for the average EhG.x; y/i � g.x; y/ the
following relation
X � �
.1/
g.x; y/ D ��xy C � g.x; p/t .x; y/Vpy C �E
N .x; y/ C
p
.2/
C � .x; y/; (4.12)
N