A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices
We present simple proofs of several basic facts of the global regime (the existence and the form of the non-random limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices, whose probability law involves the...
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509770599890944 |
|---|---|
| author | Pastur, L. A. Пастур, Л. А. |
| author_facet | Pastur, L. A. Пастур, Л. А. |
| author_sort | Pastur, L. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:55Z |
| description | We present simple proofs of several basic facts of the global regime (the existence and the form of the non-random limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices, whose probability law involves the Gaussian distribution.
The main difference with previous proofs is the systematic use of the Poincare - Nash inequality,
allowing us to obtain the O(n - 2) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter. |
| first_indexed | 2026-03-24T02:46:23Z |
| format | Article |
| fulltext |
UDC 517.4
L. A. Pastur (Math. Div. Inst. Low Temp. Phys., Kharkiv)
A SIMPLE APPROACH TO THE GLOBAL REGIME
OF GAUSSIAN ENSEMBLES OF RANDOM MATRICES
PROSTYJ PIDXID DO HLOBAL\NOHO REÛYMU
HAUSSOVYX ANSAMBLIV VYPADKOVYX MATRYC\
We present simple proofs of several basic facts of the global regime (the existence and the form of the non-
random limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of
the resolvent) for ensembles of random matrices, whose probability law involves the Gaussian distribution. The
main difference with previous proofs is the systematic use of the Poincare – Nash inequality, allowing us to
obtain the O(n−2) bounds for the variance of the normalized trace of the resolvent that are valid up to the real
axis in the spectral parameter.
Navedeno prosti dovedennq nyzky osnovnyx faktiv stosovno hlobal\noho reΩymu (isnuvannq ta vyhlqd
nevypadkovo] hranyçno] normalizovano] raxugço] miry dlq vlasnyx znaçen\, central\na hranyçna teo-
rema dlq slidu rezol\venty) dlq ansambliv vypadkovyx matryc\, do jmovirnisnoho zakonu qkyx vxodyt\
haussiv rozpodil. Holovna vidminnist\ vid poperednix doveden\ polqha[ u systematyçnomu vykorystanni
nerivnosti Puankare – Neßa, wo dozvolylo otrymaty ocinky porqdku O(n−2) dlq dyspersi] normali-
zovanoho slidu rezol\venty, qki spravdΩugt\sq do dijsno] osi vidnosno spektral\noho parametra.
1. Introduction. Numerous problems of the Random Matrix Theory can be roughly
divided in three groups or regimes, according to the order of magnitude of intervals of the
spectral axis with respect to the matrix size n. The regime in which there exists a well
defined limit of the Normalized Counting Measure (Density of States) of eigenvalues as
n → ∞ is known as global or macroscopic regime. The regime, dealing with intervals
whose length is O(n−1) (mean eigenvalue spacing) with respect to the scale, fixed by the
global regime, is known as the local or microscopic regime. This is where the repulsion
of levels, important in many applications, manifests itself. The regime in which intervals
of the length O(n−α), 0 < α < 1 are relevant is known as the intermediate. Corre-
sponding results were used in explanations of universal conductance fluctuations of small
metallic particles [1].
In obtaining results on the global and intermediate regimes the bounds of the order
o(1), n → ∞ on the variance of linear statistics
Nn[ϕ] :=
n∑
l=1
ϕ(λ(n)
l ) (1.1)
of eigenvalues
{
λ
(n)
l
}n
l=1
of random matrix in question play an important role. Most
precise bounds (up to exact asymptotic form) have the order O(n−2) and valid for suf-
ficiently smooth test functions ϕ in (1.1). These bounds, showing that eigenvalues of
random matrices are strongly dependent, appeared first in the physics literature (see e.g.
reviews [2, 3]) and were then rigorously proved for a number of random matrix ensem-
bles (see e.g. [4, 5] for ensembles with invariant probability law, and [6, 7] for the Wigner
ensembles, whose entries are independent or weakly dependent random variables modulo
symmetry conditions).
In this paper we will confine ourselves to the case ϕ(λ) = (λ − z)−1, �z �= 0 in
(1.1), corresponding to the normalized trace of the resolvent of a random matrix H
c© L. A. PASTUR, 2005
790 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 791
gn(z) := n−1Tr(H − z)−1 (1.2)
as a linear statistic.
An important ingredient of proofs in the case of invariant ensembles is an explicit
form of the joint probability law of eigenvalues (called often the Weyl formulas) and re-
lated variational and/or orthogonal polynomials techniques. As for the Wigner and other
ensembles with independent or weakly dependent entries, here the O(n−2) bounds re-
sult from an analysis of certain recurrence relations for the moments of gn. This method
is rather efficient and self-contained, but leads to O(n−2) bounds and related asymp-
totic formulas only if |�z| ≥ Cw2, where w2 is the variance of the matrix entries
{Hjk}nj,k=1 and C is an absolute constant (see e.g. [7]).
The goal of this paper is to show that if the entries are Gaussian (even dependent)
random variables, then O(n−2) bounds for the variance of ( 3.8) can be obtained by
a rather direct application of an inequality for a C1 function of a family of Gaussian
random variables. The inequality dates back to Poincare and Nash and is widely used in
statistics and analysis (see [8, 9] and references therein).
The paper is organized as follows. In Section 2 we present our technical means, in par-
ticular, an identity for expectations of differentiable functions of Gaussian random vari-
ables and the Poincare – Nash inequality. In Section 3 we find the limit of the Normalized
Counting Measure of eigenvalues for the deformed Gaussian ensembles, corresponding
to matrices that are sums of a nonrandom matrix and the matrix of the Gaussian Unitary
Ensemble (GUE) or the Gaussian Orthogonal Ensemble (GOE). An important element of
the proofs are O(n−2) bounds (3.21) and (3.40 ), proved by using the Poincare – Nash in-
equality. In Section 4 we derive the asymptotic formula for the variance of (1.2) and prove
the central limit theorem for this class of linear statistics of eigenvalues of the GUE and
the GOE by applying similar techniques. Section 5 contains a collection of related results
and outlined proofs for several other ensembles, involving Gaussian random variables:
random matrices, whose entries are dependent Gaussian random variables, the deformed
Wishart and Laguerre ensembles, ensembles, appearing in the telecommunications, and
the Wigner ensembles.
2. Technical means.
Definition 2.1 (Stieltjes transform). Let m be a finite nonnegative measure on R .
The function
f(z) =
∫
R
m(dλ)
λ− z
(2.1)
defined for all nonreal z, �z �= 0 is called the Stieltjes transform of m.
Proposition 2.1. Let f be the Stieltjes transform of a finite nonnegative measure m,
m(R) < ∞. Then:
(i) f is analytic in C \ R, and f(z) = f(z);
(ii) �f(z) · �z > 0 for �z �= 0;
(iii) limη→∞ η|f(iη)| < ∞;
(iv) for any function f, possessing the above properties there exists a finite nonneg-
ative measure m on R such that f is its Stieltjes transform;
(v) if ∆ is an interval of R whose edges are not atoms of the measure m, then we
have the Stieltjes – Perron inversion formula
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
792 L. A. PASTUR
m(∆) = lim
ε→0+
1
π
∫
∆
�f(λ + iε)dλ; (2.2)
(vi) the above one-to-one correspondence between nonnegative measures and their
Stieltjes transforms is continuous in the weak topology of measures and in the topology of
the uniform convergence on compact subsets of C \ R of analytic functions;
(vii) we have limη→∞ η|f(iη)| = m(R).
The next proposition presents elementary facts of linear algebra that will be often used
below.
Proposition 2.2. Let M be the algebra of n × n matrices with complex entries,
equipped with the Euclidean norm ‖ . . . ‖. We have:
(i) if M = {Mjk}nj,k=1 ∈ M, then
|Mjk| ≤ ‖M‖; (2.3)
(ii) if M ∈ M and TrM =
∑n
j=1
Mjj is the trace of a matrix M, then for any
M1,M2 ∈ M
|TrM1M2| ≤ (TrM1M
∗
1 )1/2(TrM2M
∗
2 )1/2, (2.4)
where M∗ is the Hermitian conjugate of M ;
(iii) if M ∈ M, then
|TrM | ≤ n‖M‖; (2.5)
(iv) for any Hermitian or real symmetric matrix M its resolvent
G(z) = (M − z)−1, G(z) = {Gjk(z)}nj,k=1 (2.6)
is defined for all nonreal z, �z �= 0, and verifies the inequalities
‖G(z)‖ ≤ |� z|−1, |Gjk(z)| ≤ |�z|−1; (2.7)
(v) if M1 and M2 are two Hermitian or real symmetric matrices and Gr(z), r =
= 1, 2, are their resolvents, then
G2(z) = G1(z) −G1(z)(M2 −M1)G2(z) (2.8)
(the resolvent identity);
(vi) if G(z) = (M−z)−1 is viewed as a function of a Hermitian or a real symmetric
matrix M, then its derivative G′(z) with respect to M verifies the relation
G′(z) ·X = −G(z)XG(z) (2.9)
for any Hermitian or real symmetric X, and
‖G′(z)‖ ≤ ‖G(z)‖2 ≤ |�z|−2. (2.10)
We present now several facts on expectations of functions of Gaussian random vari-
ables. Recall first the form of the Gaussian Orthogonal (GOE) and Gaussian Unitary
(GUE) Ensembles. These are measures, defined on the sets Mβ of n×n real symmetric
(β = 1, GOE) and (β = 2, GUE) Hermitian matrices M = {Mjk}nj,k=1 respectively,
and given by
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 793
Pβ(dβM) =
1
Zn,β
exp
(
− nβ
4w2
TrM2
)
dβM, β = 1, 2, (2.11)
where Zn,β is a normalizing constant and
d1M =
∏
1≤j≤k≤n
dMj,k, d2M =
n∏
j=1
dMj,j
∏
1≤j<k≤n
d�Mj,kd�Mj,k. (2.12)
Proposition 2.3. Consider the GOE (the GUE) and let Φ : Mn → C be a C1
function, bounded together with its derivative. Then for any (real symmetric) Hermitian
matrix X we have
E{Φ′(M) ·X} =
βn
2w2
E
{
Φ(M)Tr(MX)
}
, (2.13)
where the symbol E{. . .} denotes the expectation with respect to the GOE (β = 1) and
the GUE (β = 2) measures (2.11).
Proof. Consider the integral
I =
∫
Mβ
Φ(M) exp{−βnTrM2/4w2}dβM.
Since the measures dβM, β = 1, 2, are invariant with respect to translations M →
→ M + εX for any X ∈ Mβ and ε ∈ R, we have
I =
∫
Mβ
Φ(M + εX) exp{−βnTr(M + εX)2/4w2}dβM.
Differentiating this expression with respect to ε and then setting ε = 0, we obtain the
assertion.
Remarks. 2.1. Taking the case n = 1, β = 1, in the proposition and denoting
2w2 = σ2 we obtain
1√
2πσ2
∫
R
xΦ(x)e−x
2/2σ2
dx = σ2 1√
2πσ2
∫
R
Φ′(x)e−x
2/2σ2
dx (2.14)
or
E{ξΦ(ξ)} = E{ξ2}E{Φ′(ξ)}, (2.15)
where ξ is the Gaussian random variable of zero mean and of variance σ2. The first for-
mula shows that the proposition is a matrix version of the integration by parts. The second
formula makes explicit the ”decoupling” nature of (2.13), whose analogs are widely used
in various domains of mathematical physics.
2.2. It is easy to prove a multivariate version of (2.15). Namely, if X = {ξj}qj=1 ∈
∈ R
q is a random Gaussian vector such that
E{ξj} = 0, E{ξjξk} = Cjk, j, k = 1, . . . , q, (2.16)
and Φ: R
q → C has bounded partial derivatives, then
E{ξjΦ} =
q∑
k=1
CjkE{(�Φ)k}, (�Φ)k =
∂Φ
∂xk
. (2.17)
Next result is known as the Poincare – Nash inequality (see e.g. [8, 9] and references
therein) and will also play an important role below.
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794 L. A. PASTUR
Proposition 2.4. Consider a random Gaussian vector X = {ξj}pj=1, satisfying
(2.16) with p = q, and Φ1,2 : R
p −→ C, having bounded partial derivatives. Then
Cov{Φ1,Φ2} := E {Φ1Φ2)} − E{Φ1}E{Φ2} ≤
≤ E
{
(C�Φ1,�Φ1 )
}1/2
E(C�Φ2,�Φ2 )1/2, (2.18)
where
(C�Φ,�Φ) :=
p∑
j,k=1
Cjk(�Φ)j(�Φ)k. (2.19)
In particular, if Φ: R
p −→ C has bounded partial derivatives, then
Var{Φ} := E
{
|Φ|2)
}
− |E{Φ}|2 ≤ E {(C�Φ,�Φ)} . (2.20)
Proof. We will outline a proof, based on (2.17). Consider two q-component inde-
pendent Gaussian vectors X(1) and X(2) with zero means and the covariance matrices
C(1) and C(2). Define the “interpolating” Gaussian vector
X(t) =
√
tX(1) +
√
1 − tX(2), t ∈ [0, 1]. (2.21)
Then for any Ψ: R
q −→ C with bounded first and second partial derivatives we have
E
{
Ψ(X(1))
}
− E
{
Ψ(X(2))
}
=
=
1
2
1∫
0
E
{(
(C(1)�Ψ,�Ψ) − (C(2)�Ψ,�Ψ)
)
(X(t))
}
dt. (2.22)
Indeed, write the l.h.s. of (2.22) as
1∫
0
d
dt
E {Ψ(X(t)} dt =
=
1∫
0
E
{(((
2
√
t
)−1
X(1) −
(
2
√
1 − t
)−1
X(2)
)
,�Ψ
(
X(t)
))}
dt.
Now, by using (2.17) in each term of the r.h.s., we obtain (2.22).
To prove (2.18), (2.19), we choose
X(1) = (X ′, Y ′), where (X ′, Y ′) is the q = 2p-component Gaussian vector, whose
distribution is concentrated on the “diagonal” X ′ = Y ′ and has there zero mean and the
covariance matrix C,
X(2) = (X ′′, Y ′′), where X ′′ and Y ′′ are independent p-component Gaussian
vectors of zero mean and of covariance matrix C,
Ψ(X,Y ) = Φ1(X)Φ2(Y ).
In other words, X(1) and X(2) are q = 2p-component Gaussian vectors with zero mean
and with covariance matrices
C(1) =
(
C C
C C
)
, C(2) =
(
C 0
0 C
)
.
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 795
It is easy to see that for this choice of C(1) and C(1) the covariance Cov{Φ1,Φ2} has
the form of the l.h.s. of (2.22), and we obtain
Cov{Φ1,Φ2} =
1∫
0
E
{
(C�Φ1(X̂(t)),�Φ2(Ŷ (t)))
}
dt,
where X̂(t) =
√
tX ′ +
√
1 − tX ′′, Ŷ (t) =
√
tY ′ +
√
1 − tY ′′, t ∈ [0, 1]. Now,
to obtain (2.18), we use Schwarz inequality |(CX, Y )|2 ≤ (CX,X)(CY, Y ), valid for
a positive definite matrix C and any two vectors X,Y ∈ C
p, Schwarz inequality for
mathematical expectations, and the fact that X̂(t) and Ŷ (t) are identically distributed
Gaussian vectors, whose common law is determined by the matrix C, hence does not
depend on t.
3. Deformed semicircle law. Denote
{
λ
(n)
l
}n
l=1
eigenvalues of a n× n real sym-
metric or Hermitian matrix H and introduce the Normalized Counting Measure of eigen-
values (NCM)
Nn(∆) =
1
n
n∑
l=1
χ∆
(
λ
(n)
l
)
, (3.1)
where χ∆ is the indicator of an interval ∆ ⊂ R. The NCM is a particular case of linear
statistics (1.1), corresponding to ϕ = χ∆.
We will consider in this section the convergence of the Normalized Counting Mea-
sures of eigenvalues of the Gaussian ensembles, a basic result of the global regime, perti-
nent for any subsequent study of the eigenvalue distribution of random matrix in question.
In particular, we are going to prove that NCM converges with probability 1 to a nonran-
dom measure, known as the deformed semicircle or the Wigner law.
We begin with Hermitian n× n matrices and consider the ensemble of the form
H = H(0) + M, (3.2)
where H(0) is a nonrandom Hermitian matrix and M is a random matrix, distributed
according to the GUE law, defined by (2.11), (2.12) with β = 2. Random matrices of
this form can be viewed as “perturbations” or “deformations” of the GUE matrix by a
nonrandom matrix H(0). We will call (3.2) the deformed GUE.
Writing
M = W/n1/2, (3.3)
we find from (2.11) that the entries Wjj , j = 1, . . . , n, �Wj,k, and �Mj,k, 1 ≤ j <
< k ≤ n, are independent Gaussian random variables, defined by the equalities:
E{Wjk} = 0, E{W 2
jk} = 0, E{|Wjk|2} = w2(1 + δjk)/2. (3.4)
This shows that the random variables {Wn
jk}nj,k=1 can be viewed as the upper left corner
of the semiinfinite Hermitian matrix {Wn
jk}∞j,k=1, whose entries are complex Gaussian
random variables, defined by (3.4) for 1 ≤ j, k ≤ ∞. This observation will allow us to
use the convergence with probability 1 in the probability space, defined by {Wn
jk}∞j,k=1.
Theorem 3.1 (deformed semicircle law). Given n ∈ N, consider the deformed
Gaussian Unitary Ensemble (3.2) of n×n random Hermitian matrices, defined by (3.2) –
(3.4). Assume that the Normalized Counting Measure of eigenvalues N
(0)
n of H(0) con-
verges weakly to a nonnegative unit measure N (0) and denote
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
796 L. A. PASTUR
f (0)(z) =
∫
N (0)(dλ)
λ− z
, �z �= 0, (3.5)
be the Stieltjes transform of N (0). Let Nn be the Normalized Counting Measure of the
ensemble. Then there exists a nonnegative unit measure Ndsc such that with probability
1 we have the weak convergence:
limn→∞Nn = Ndsc, (3.6)
and the Stieltjes transform fdsc of Ndsc is a unique solution of the functional equation
f(z) = f (0)(z + w2f(z)), (3.7)
in the class of functions, analytic for �z �= 0 and such that �f(z) · �z > 0.
In view of the one-to-one correspondence between measures and their Stieltjes trans-
forms (see Proposition 2.1) it suffices to study the Stieltjes transform
gn(z) =
∫
Nn(dλ)
λ− z
, �z �= 0, (3.8)
of the Normalized Counting Measure Nn. The spectral theorem for Hermitian matrices
implies the formula
gn(z) = n−1TrG(z), (3.9)
where
G(z) = (H − z)−1 (3.10)
is the resolvent of H (see Proposition 2.2 (iv)). This link between the NCM of a Hermi-
tian (real symmetric) matrix and its resolvent will play an important role in what follows.
In particular, it motivates the next lemma.
Lemma 3.1. Let G(z) = (H−z)−1 be the resolvent of the matrix (3.2), G(0)(z) be
the resolvent of H(0), and gn be defined by (3.8), (3.9). Then we have for any nonreal z
E{G(z)} = G(0)(z̃(z)) + w2E
{◦
gn(z)G(z)
}
G(0)(z̃(z)), (3.11)
where
◦
gn(z) = gn(z) − fn(z), fn(z) = E{gn(z)}, (3.12)
and
z̃ = z + w2fn(z). (3.13)
Proof. Let j, k be two indexes varying between 1 and n. Applying Proposition 2.3
with β = 2 to the function Φ(M) = (H(0) + M − z)−1
jk := Gjk, and using (2.9), we
obtain for any Hermitian matrix X
E{(GXG)jk} +
n
w2
E{GjkTrMX} = 0. (3.14)
We choose here X as
X =
{
X
(p,q)
jk
}n
p,q=1
, X
(p,q)
jk = aδjpδkq + aδjqδkp, a ∈ C, (3.15)
where p and q are two given indexes, varying between 1 and n. This yields
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 797
E{GjkMqp} = −w2
n
E{GjpGqk}, (3.16)
where {Mjk}nj,k=1 are the entries of M of (3.2). By using the resolvent identity (2.8)
for the pair (H,H(0))
G = G(0) −GMG(0), (3.17)
we can write the equality
E{Gjk} = G
(0)
jk −
n∑
p,q=1
E{GjqMqp}G(0)
pk . (3.18)
Replacing the expectation in the sum by the r.h.s. of (3.16) with k = q, and using
notation (3.12), (3.13), we obtain the following matrix form of the previous relation:
E{G(z)}
(
1 − w2fn(z)G(0)(z)
)
= G(0)(z) + w2E{◦gn(z)G(z)}G(0)(z). (3.19)
We have also:
1 − w2fn(z)G(0)(z) = (H(0) − z − w2fn(z))G(0)(z). (3.20)
It follows from (3.12) that fn is the Stieltjes transform of a probability measure, and by
(2.7) we have for �z �= 0 : |�(z + w2f(z))| > |�z| > 0. Hence the matrix H(0) −
−z − w2fn(z) is invertible uniformly in n if �z �= 0. Its inverse is G(0)(z̃ ), and we
can write the r.h.s. of (3.20) as (G(0)(z̃ ))−1G(0)(z). This and (3.19) imply (3.11).
Theorem 3.2. Let gn(z) be as in (3.9), (3.10), where H is given by (3.2). Then
Var{gn(z)} := E
{
|gn(z) − E{gn(z)}|2
}
≤ w2
n2|�z|4 . (3.21)
Proof. We will use inequality (2.20), choosing the GUE matrix M as X and gn(z)
as Φ. We have by (2.9):
∂gn(z)
∂Mjj
= − 1
n
(G2)jj ,
∂gn(z)
∂�Mjk
= − 1
n
[
(G2)jk + (G2)kj
]
,
∂gn(z)
∂�Mjk
= − i
n
[
(G2)jk − (G2)kj
]
.
According to (2.11) with β = 2 Mjj , j = 1, . . . , n, �Mjk, 1 ≤ j < k ≤ n, �Mjk,
1 ≤ j < k ≤ n, are independent Gaussian random variables of zero mean and of variance
E{Mjj
2} =
w2
n
, E{(�Mjk)2} = E{(�Mjk)2} =
w2
2n
. (3.22)
Hence, the r.h.s. of (2.20) will be in this case:
E
w2
n3
n∑
j=1
∣∣(G2)jj
∣∣2 +
w2
2n3
∑
1≤j<k≤n
∣∣(G2)jk + (G2)kj
∣∣2 +
∣∣(G2)jk − (G2)kj
∣∣2 =
=
w2
n3
E
n∑
j,k=1
∣∣(G2)jk
∣∣2 =
w2
n3
E
{
TrG2(z)G2(z∗)
}
.
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798 L. A. PASTUR
In view of (2.7) the r.h.s. admits the bound w2/n2|�z|4, coinciding with the r.h.s.
of (3.21).
Proof of Theorem 3.1. According to (3.8) gn(z) = n−1 TrG(z) is the Stieltjes
transform of the Normalized Counting Measure Nn. By applying the operation n−1 Tr
to formula (3.11), we obtain for fn(z) = E{gn(z)} :
fn(z) = f (0)
n (z + w2fn(z)) + w2E
{◦
gn(z)n−1TrG(z)G(0)(z̃ )
}
, (3.23)
where
f (0)
n (z) =
∫
N
(0)
n (dλ)
λ− z
, �z �= 0. (3.24)
By using (3.21), (2.7), and Schwarz inequality, we estimate the second term in the r.h.s. of
(3.23) by the expression w2|�z|−2E
{
|◦gn(z)|2
}1/2
, bounded from above by w3/n|�z|4
in view of Theorem 3.2. We obtain the inequality∣∣∣fn(z) − f (0)
n (z + w2fn(z))
∣∣∣ ≤ w3
n|�z|4 . (3.25)
In view of (2.7) the sequence {fn} consists of functions, analytic and uniformly bounded
in n and in z by η−1
0 < ∞ if |�z| ≥ η0 > 0. Hence, there exists a function f and
an infinite subsequence {fnj}j≥1 that converges to f uniformly on any compact set of
C \ R. According to Proposition 2.1 (iii) we have
�fn(z) · �z > 0, �z �= 0, (3.26)
thus �f(z) ·�z ≥ 0, �z �= 0. In addition, according to Proposition 2.1 and the hypoth-
esis of the theorem on the convergence of the sequence {N (0)
n } to N (0), the sequence
{f (0)
n } of (3.24) is analytic in C \R and converges uniformly on compact sets of C \R
to the Stieltjes transform f (0) of the limiting counting measure N (0) of “unperturbed”
matrices H(0). This allows us to pass to the limit nj → ∞ in (3.25) and to obtain that
the limit of any converging subsequence of the sequence {fn} satisfies the functional
equation (3.7). According to Lemma 3.2 below, the equation is uniquely soluble in the
class of functions, analytic for �z �= 0 and such that �f(z) · �z ≥ 0, �z �= 0, and the
solution possesses the property �f(z) · �z > 0, �z �= 0. Hence the whole sequence
{fn} converges to the unique solution fdsc of the equation.
In addition, the Tchebyshev inequality and Theorem 3.2 imply that for any ε > 0,
P{|fn(z) − gn(z)| > ε} ≤ 1
ε2
Var{gn(z)} ≤ w2
ε2|�z|4n2
.
Hence the series
∑∞
n=1
P{|fn(z)− gn(z)| > ε} converges for any ε > 0, and |�z| ≥
≥ η0 > 0, and by the Borel – Cantelli lemma we have for any fixed z, |�z| ≥ η0 >
> 0 with probability 1 limn→∞ gn(z) = f(z). Let us show that gn converges to f
uniformly on any compact of C \ R with probability 1. Because of the uniqueness
of analytic continuation it suffices to prove that with the same probability the limiting
relation limn→∞ gn(zj) = f(zj) is valid for all points of an infinite sequence {zj}j≥1,
zj , |�zj | ≥ η0 > 0, possessing an accumulation point. Indeed, according to the above
P{Ω(zj} = 1 ∀j. Hence
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 799
P
⋂
j≥1
Ω(zj)
= 1,
and the last assertion is proved.
Denote by Ndsc the nonnegative measure, whose Stieltjes transform is fdsc. Then,
by Proposition 2.1(vi), we have with probability 1 the weak convergence (3.6), and, in
view of Lemma 3.2, Ndsc is a unit measure. The measure can be found by using the
inversion formula of Proposition 2.1(v).
Remark 3.1. If in the conditions of the above theorem we assume additionally that
the sequence {N (0)
n } is tight, then the sequence {Nn} is also tight with probability 1.
Indeed, consider first the case of the GUE itself, corresponding to H(0) = 0 in (3.2). In
this case we have by definition of the NCM and by (3.3)∫
λ2Nn(dλ) =
1
n2
n∑
j,k=1
|Wjk|2. (3.27)
It is easy to prove that the sum on the r.h.s. of (3.27) converges with probability 1 to
E{|W12|2} = w2
(
this is the strong law of large numbers for the Gaussian random
variables {Wjk}∞j,k=1
)
. Hence, the second moment of Nn is bounded uniformly in n
with probability 1 and the sequence {Nn} is tight with probability 1.
In a general case of the deformed GUE (3.2), (3.3) we can argue as follows. We first
use the resolvent identity (3.17) and inequalities (2.4) – (2.7), according to which
|gn(z) − g(0)
n (z)| ≤
∣∣∣n−1TrG(z)MG(0)(z)
∣∣∣ ≤ 1
|�z|2
n−2
n∑
j,k=1
|Wjk|2
1/2
.
Next, we note that if m is a unit nonnegative measure and f is its Stieltjes transform
then
−(1 + η)−1 + m({λ : |λ| ≤ η1/2}) ≤
≤ η�f(iη) ≤ η|f(iη)| ≤ m({λ : |λ| ≤ η3/2}) + (1 + η)−1/2.
By using these inequalities we obtain the bounds
Nn({λ : |λ| ≤ η3/2}) ≥
≥ N (0)
n ({λ : |λ| ≤ η1/2}) − (1 + η)−1 − (1 + η)−1/2 −W 1/2
n η−1,
where Wn is the r.h.s. of (3.27). Since, according to the above, Wn is bounded with
probability 1, the bound and the tightness of the sequence
{
N
(0)
n
}
implies the tightness
of {Nn}.
Lemma 3.2. Let f (0) be the Stieltjes transform of a unit nonnegative measure, and
w be a positive number. Then the functional equation
f(z) = f (0)(z + w2f(z)) (3.28)
has at most one solution, analytic for �z �= 0 and such that
�f · �z ≥ 0. (3.29)
The solution is the Stieltjes transform of a unit nonnegative measure N, in particular,
inequality (3.29) is strict: �f · �z > 0.
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800 L. A. PASTUR
Proof. Let us prove first that for any solution of (3.28), (3.29) the inequality (3.29)
is strict. Assume that �f(z0) = 0 for some z0, �z0 �= 0. Then (3.28) implies that
�f (0)(z0) = 0. This is impossible because, according to (3.5),
�f (0)(z) = �z
∫
N (0)(dµ)
|µ− z|2
is strictly positive for any nonreal z, and any nonnegative unit measure N (0).
Let us prove now that (3.28) is uniquely soluble. Assume that (3.28), (3.29) possesses
two solutions f1 and f2. Since in any bounded part of the upper half-plane they coincide
at most at a finite number of points, there exists a subsequence {zp}, such that zp → ∞
as p → ∞ and f1(zp) �= f2(zp) ∀p. By using (3.28) and (3.5), we obtain the relation
1 = w2
∫
N (0)(dµ)
(µ− zp − w2f1(zp)) (µ− zp − w2f2(zp))
that has to be valid for all zp. This is impossible, because the limit of the r.h.s. of the
relation is zero as p → ∞ (recall that |f1,2(z)| ≤ |�z|−1 ).
To prove that the solution of (3.28), (3.29) is the Stieltjes transform of a unit
nonnegative measure, we have to prove that limη→∞ η|f(iη)| = 1 (see Proposi-
tion 2.1 (vii)). Since f (0) possesses the same property, it suffices to prove the equality
limη→∞ η|f(iη)| = limη→∞ η|f (0)(iη)|. It follows readily from (3.28) and the inequal-
ity � z̃ ≥ �z.
Corollary 3.1. Consider the GUE, and let Nn be its Normalized Counting Measure
of eigenvalues. Then there exists a unit measure Nsc, called the semicircle law and such
that the sequence {Nn} converges tightly to Nsc with probability 1 :
limn→∞Nn = Nsc,
and
Nsc(∆) =
∫
∆
ρsc(λ)dλ, ρsc(λ) =
1
2πw2
(4w2 − λ2)1/2+ , (3.30)
where we denote here and below
x+ = max(x, 0), x ∈ R. (3.31)
Proof. The case of the GUE corresponds to H(0) = 0 in (3.2). The normalizing
counting measure of this matrix is the unit measure, concentrated at 0, and its Stieltjes
transform is f
(0)
n (z) = −1/z. Its limit is the same, hence equation (3.28) in this case is
f(z) = − 1
z + w2f(z)
, (3.32)
or
w2f2(z) + zf(z) + 1 = 0. (3.33)
A solution of this quadratic equation that satisfies the condition �f(z) · �z ≥ 0, �z �= 0
is unique and is given by
f(z) =
1
2w2
(√
z2 − 4w2 − z
)
, (3.34)
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 801
where
√
z2 − 4w2 denotes the branch that has the asymptotic behavior
√
z2 − 4w2 =
= z + O(|z|−1), z → ∞. In particular, this branch assumes purely imaginary values
with positive imaginary part on the upper edge of the cut (−2w, 2w). Applying to (3.34)
the inversion formula (2.2), we obtain (3.30).
Remarks. 3.2. The case of the GUE itself requires fewer technicalities, than the gen-
eral case of matrices (3.2). Indeed, since in this case G(0) = −z−1, the operation
n−1 Tr, applied to (3.11) with this G(0), yields
w2f2
n(z) + zfn(z) + 1 = −w2E{◦g
2
n(z)}. (3.35)
Hence, Theorem 3.2 leads directly to the quadratic equation (3.33). The unique solubility
of this equation in the class of analytic functions verifying (3.29) is immediate.
3.3. For the deformed Gaussian Orthogonal Ensemble the limiting NCM is the same,
i.e., it is given by the deformed semicircle law, although the proof is more involved. We
outline the proof in the case of the GOE itself, indicating only moments that are different
from those of the proof of Theorem 3.1.
Recall first that according to (2.11) the GOE corresponds to n × n real symmetric
matrices of the form (cf. (3.3))
M = W/n1/2, (3.36)
where W = {Wjk} are Gaussian random variables, independent for 1 ≤ j < k ≤ n
and such that (cf. (3.4))
Wjk = Wkj , E{Wjk} = 0, E{W 2
jk} = w2(1 + δjk). (3.37)
Hence we can again view {Wjk}nj,k=1 as the n× n upper left corner of the semiinfinite
real symmetric matrix {Wjk}∞j,k=1 with Gaussian entries, defined by (3.37), and we can
consider the convergence with probability 1 in the corresponding probability space.
By using Proposition 2.3 for the case β = 1 of real symmetric matrices and choosing
as X the matrices X(p,q) = {X(p,q)
jk }nj,k=1, p, q = 1, . . . , n, with X
(p,q)
jk = δjpδkq +
+ δjqδkp (cf. (3.15)), we obtain instead of (3.16) the relation
E{GjkMqp} = −w2
n
E{GjpGqk} −
w2
n
E{GjqGpk}, (3.38)
valid for j, k, p, q = 1, . . . , n, and containing the additional cross term in the r.h.s. This
leads to the following analog of (3.35):
w2f2
n(z) + zfn(z) + 1 = −w2E{(◦gn(z))2} − n−2w2E{TrG2(z)}, (3.39)
containing the term w2n−2E{TrG2(z)}, absent in the GUE case (see (3.35)). In view
of (2.5) the term admits the bound |w2n−2E{TrG2(z)}| ≤ w2/n|�z|2, hence does not
contribute to the limiting form of (3.39). The form coincides with (3.33) provided that
the variance E
{
|◦gn(z)|2
}
vanishes as n → ∞. This fact, namely an analog of bound
(3.21) for the GOE, can be proved by the same argument as in the case of the GUE above.
Indeed, applying again inequality (2.20) to gn(z), we obtain
Var{gn(z)} ≤ 2w2
n2|�z|4 . (3.40)
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802 L. A. PASTUR
4. Variance and central limit theorem for the trace of the resolvent.
4.1. Variance. Next theorem is a more detailed version of Theorem 3.2. To avoid
technicalities, we confine themselves the case of the GUE itself.
Theorem 4.1. Consider the GUE. Let gn(z) be defined by (3.8) – (3.10) with H(0) =
= 0. Then we have for n → ∞
Cov{gn(z1), gn(z2)} = d2(z1, z2)n−2 + r(2)
n (z1, z2), (4.1)
where
d2(z1, z2) = − 1
2(z1 − z2)2
(
1 − z1z2 − 4w2√
z2
1 − 4w2
√
z2
2 − 4w2
)
, (4.2)
and r
(2)
n admits the bound
|r(2)
n (z1, z2)| ≤ C/n3, (4.3)
where C is independent on n and finite if min{|�z1|, |�z2|} > 0.
Proof. We can write by definition
Cov{gn(z1, z2)} = E
{
gn(z1)
◦
gn(z2)
}
. (4.4)
Applying Proposition 2.3 to the r.h.s., we obtain the identity
E{gn(z1)
◦
gn(z2)} = −w2
z1
E{g2
n(z1)
◦
gn(z2)} −
w2
z1n3
E{TrG(z1)G2(z2)}. (4.5)
This, (3.12), and the relations
G(z1)G(z2)} =
G(z1) −G(z2)}
z1 − z2
,
d
dz
G(z) = G2(z), (4.6)
allow us to rewrite the r.h.s. of the identity as
−2w2
z1
fn(z1)E
{
gn(z1)
◦
gn(z2)
}
− w2
z1n2
∂
∂z2
fn(z1) − fn(z2)
z1 − z2
− w2
z1
E
{◦
g
2
n(z1)
◦
gn(z2)
}
.
Hence, we obtain from (4.4)
Cov {gn(z1), gn(z2)} =
= − w2
z1 + 2w2fn(z1)
{
1
n2
∂
∂z2
fn(z1) − fn(z2)
z1 − z2
+ E{◦g
2
n(z1)
◦
gn(z2)}
}
, (4.7)
where z1 + 2w2fn(z1) �= 0 if |�z1| �= 0 because �fn(z) · �z > 0 for �z �= 0.
Moreover, we have the bound
|z + 2w2fn(z)| ≥ |�(z + 2w2fn(z))| ≥ |�z|. (4.8)
Consider the contribution of the first term of the r.h.s. of (4.7). By (3.35), (3.33) and
Theorem 3.2 we have
|f(z) − fn(z)| ≤ w2
z + w2|f(z) + fn(z)|Var{gn(z)} ≤ w4
n2|�z|5 ,
where we took into account (3.26) and the inequality �f(z) · �z > 0 for �z �= 0,
implying that
∣∣z + w2(f(z) + fn(z))
∣∣ ≥ |�z|.
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Besides, since fn is analytic for �z �= 0, we have for |�z1,2| ≥ η0 > 0 :
∂
∂z2
fn(z1) − fn(z2)
z1 − z2
=
1∫
0
f ′′
n (z1 + t(z2 − z1))tdt,
and
f ′′
n (z) =
1
πi
∫
|ζ−z|=η0/2
fn(ζ)dζ
(ζ − z)3
. (4.9)
The above three relations imply that the replacement fn by f in (4.7) yields an error
term bounded by C(η0)/n4, where C(η0) is finite if η0 > 0.
We have then
Cov{gn(z1), gn(z2)} = − w2
n2(z1 + 2w2f(z1))
∂
∂z2
f(z1) − f(z2)
z1 − z2
−
− w2
z1 + 2w2fn(z1)
E
{
◦
g
2
n(z1)
◦
gn(z2)
}
+ O(n−4), (4.10)
if min{|�z1|, |�z2|} > 0. Now it is easy to show by using (3.34) that the first term in
the r.h.s. coincides with the first term of the r.h.s. of (4.1).
To finish the proof we have to show that E
{◦
g2
n(z1)
◦
gn(z2)
}
is of the order O(n−3).
Indeed, by Schwarz inequality∣∣∣E{◦
g2
n(z1)
◦
gn(z2)
}∣∣∣ ≤ Var1/2
{◦
g2
n(z1)
}
Var1/2{gn(z2)}.
The second factor of the r.h.s. is estimated in Theorem 3.2. To estimate the first term we
use again the Poincare – Nash bound (2.18), (2.19). This and Theorem 3.2 yield
Var
{◦
g
2
n(z1)
}
≤ 4w2
n3
E
{∣∣∣◦gn(z1)
∣∣∣2 TrG2(z1)G2(z∗1)
}
≤ 4w4
n4|�z1|8
, (4.11)
and we obtain the inequality
∣∣∣E{◦
g
2
n(z1)
◦
gn(z2)
}∣∣∣ ≤ 2w3
/
n3|�z1|4|�z2|2. This proves
bound (4.3), hence the theorem.
Remarks. 4.1. Similar argument shows that in the case of the GOE we have
Cov{gn(z1), gn(z2)} = d1(z1, z2)n−2 + r(1)
n (z1, z2), (4.12)
where
d1(z1, z2) := − 1
(z1 − z2)2
(
1 − z1z2 − 4w2√
z2
1 − 4w2
√
z2
2 − 4w2
)
, (4.13)
and r
(1)
n admits the same bound as (4.3).
4.2. It is convenient to write a unique formula for (4.2) and (4.13):
dβ(z1, z2) := − 1
β(z1 − z2)2
(
1 − z1z2 − 4w2√
z2
1 − 4w2
√
z2
2 − 4w2
)
, β = 1, 2. (4.14)
4.3. We mention also another expression for dβ(z1, z2) :
dβ(z1, z2) =
2w2
β(1 − w2f2(z1))(1 − w2f2(z2))
(
f(z1) − f(z2)
z1 − z2
)2
, (4.15)
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804 L. A. PASTUR
where f is the Stieltjes transform (3.34) of the semicircle law, the limiting Normalized
Counting Measure of eigenvalues for the GUE and the GOE.
4.4. According to physics literature (see e.g. [3]) the expectation of any unitary in-
variant and smooth function of the GUE matrix admits an expansion in n−2. Since
◦
gn(z1)
◦
gn(z2) is smooth and unitary invariant, we have to expect in this case that the
error term in (4.2) is of the order O(n−4). This requires a bound O(n−4) for the second
term of the r.h.s. of (4.10) that can be proved as follows. By repeating the argument that
led from (4.5) to (4.10), we obtain
E
{◦
g2
n(z1)
◦
gn(z2)
}
=
= − w2
z1 + 2w2fn(z1)
E
{(◦
g2
n(z1) − E
{◦
g2
n(z1)
})◦
gn(z1)
◦
gn(z2)
}
+ O(n−4.).
The expectation in the r.h.s. is estimated by
(
Var
{◦
g2
n(z1)
}
Var
{◦
gn(z1)
◦
gn(z2)
})1/2
.
Now, by applying (2.20), we find that the both variances are of the order O(n−4) (cf.
(4.11)), hence this expression is O(n−4) as well. This implies the same order of magni-
tude of the error term in (4.2).
4.2. Central limit theorem. The results of Section 3 can be viewed as an analog of
the strong law of large numbers for the linear statistics (1.2). In this section we consider
the central limit theorem for these linear statistics.
According to (3.21) and (3.40) the variance of gn = n−1 Tr(M − z)−1, the normal-
ized trace of the resolvent of M of (3.3), (3.4), is of the order O(n−2) for the Gaussian
ensembles. Hence the central limit theorem should be valid for the trace itself
γn(z) :=
n∑
l=0
1
λl − z
= ngn(z). (4.16)
This has to be compared with the case of the i.i.d. random variables with the finite second
moment, where the variance of linear statistics is always of the order O(n−1), and the
central limit theorem is valid for linear statistics multiplied by
√
n.
We will begin from the technically simplest case of the random variable
γR,n(z) := �γn(z), (4.17)
and the Gaussian Unitary Ensemble.
Theorem 4.2. Consider the GUE. Then for any fixed z such that |�z| �= 0 the
random variable
◦
γR,n(z) := γR,n(z) − E{γR,n(z)} (4.18)
converges in distribution to the Gaussian random variable
◦
γR(z) whose expectation is
zero and whose variance is
v2(z, z ) := Var{γR(z)} =
1
4
(
d2(z, z) + d2(z, z ) + 2d2(z, z )
)
, (4.19)
where d2(z1, z2) is given by (4.2).
Proof. Consider the characteristic function of
◦
γR,n(z)
Fn(t) := E
{
exp{it◦γR,n(z)}
}
, t ∈ R.
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It suffices to prove that for any fixed t ∈ R Fn(t) converges to e−v2t
2/2 as n → ∞,
where v2 is given by (4.19).
We have evidently
d
dt
Fn(t) =
i
2
[
An(t) + Bn(t)
]
, (4.20)
where
An(t) := E
{◦
γn(z) exp{it◦γR,n(z)}
}
, Bn(t) := E
{ ◦
γn(z) exp{it◦γR,n(z)}
}
,
(4.21)
and
◦
γn(z) := γn(z) − E{γn(z)} (4.22)
is the centralized trace of the resolvent (cf. (4.18)). Applying Proposition 2.3 to Φ =
= Gjk exp
{
it
◦
γR,n
}
and performing simple transformations, we obtain
n[1 + zfn(z)]Fn(t) + zAn(t) + w2E
{
n−1γ2
n(z) exp{it◦γR,n(z)}
}
+
+
it
2
w2E
{[
n−1TrG3(z) + n−1TrG2(z)G(z)
]
exp{it◦γR,n(z)}
}
= 0,
where fn(z) = E{n−1TrG(z)} (see (3.12)) and we took into account that γn(z) =
= γn(z ). Set here t = 0, multiply the result by Fn(t), taking into account that
Fn(0) = 1, An(0) = 0, and subtract the obtained equality from the above. This and the
identity
γ2
n − E{γ2
n(z)} =
◦
γ2
n − E
{◦
γ2
R(z)
}
+ 2
◦
γnnfn
yield
(z + 2w2fn(z))An(t) =
= −w2E
{
n−1 ◦
γ2
n(z) exp{it◦γR,n(z)}
}
+ w2E
{
n−1 ◦
γ2
n(z)
}
Fn(t)−
− it
2
w2E
{[
n−1TrG3(z) + n−1TrG2(z)G(z)
]
exp{it◦γR,n(z)
}
. (4.23)
According to Theorem 3.2, we have
|E{n−1(
◦
γn(z))2}| ≤ nVar{gn(z)} ≤ w2
n|�z|2
(recall that according to (3.8) gn(z) := n−1TrG(z) = n−1γn(z) ). Hence, the first and
the second terms of the r.h.s. of (4.23) vanish as n → ∞. This, (4.6), and the obvious
formula G3(z) = G′′(z)/2 allow us to write (4.23) in the following asymptotic form (cf.
the r.h.s. of (4.7)):
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806 L. A. PASTUR
(z + 2w2fn(z))An(t) =
= − it
2
w2E
{[
1
2
d2
dz2
gn(z) +
∂
∂z
gn(z) − gn(z )
z − z
]
exp{it◦γR,n(z)}
}
+ o(1). (4.24)
Since gn(z) is analytic for |�z| �= 0, we have by the Cauchy formula for any |�z| ≥
≥ η0 > 0
d2
dz2
◦
gn(z) =
1
πi
∫
|ζ−z|=η0/2
◦
gn(ζ)dζ
(ζ − z)3
.
This, (3.21), and Schwarz inequality imply
Var
{
d2
dz2
gn(z)
}
≤ Aw2
n2η8
0
, (4.25)
where A is an absolute constant. This estimate, (4.8), and Corollary 3.1, according to
which the limit limn→∞ fn(z) = f(z) is uniform on a compact set of C \ R, allow us
to replace (4.24) by
An(t) = itL(z, z)Fn(t) + o(1), n → ∞,
where
L(z, z ) = − w2
2(z + 2w2f(z))
(
1
2
d2f(z)
dz2
+
∂
∂z
f(z) − f(z )
z − z
)
, (4.26)
and f is given by (3.34).
Similar argument leads to the asymptotic relation for the second term Bn(t) of (4.20),
defined in (4.21):
Bn(t) = itL(z, z )Fn(t) + o(1), |�z| ≥ η0 > 0, n → ∞.
We obtain now from (4.20):
d
dt
Fn(t) = −tv2(z, z )Fn(t) + rn(t, z, z ), (4.27)
where
v2(z, z ) = �L(z, z ), (4.28)
and
lim
n→∞
rn(t, z, z ) = 0 (4.29)
uniformly in |�z| ≥ η0 > 0 and in t, varying in any finite interval.
Since Fn(0) = 1, we can write (4.27) as
Fn(t) = e−
v2t2
2 +
t∫
0
e−
v2
2 (t2−s2)rn(s, z, z )ds,
implying, together with (4.29), that uniformly in |�z| ≥ η0 > 0 and in t, varying in any
finite interval,
lim
n→∞
Fn(t) = exp
{
−v2(z, z )t2
2
}
.
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 807
This and the standard continuity theorem for characteristic functions proves that the ran-
dom variable
◦
γR,n(z) converges in distribution to a Gaussian random variable
◦
γR(z) of
zero mean and of variance v2(z, z) given by (4.28). In view of (4.26), (3.34), and (4.19)
we obtain the expression (4.2) for the variance after a simple algebra.
Remarks. 4.5. According to Theorem 4.1 d2(z1, z2) is the leading term of the co-
variance of γn(z1) and γn(z2) as n → ∞. This is in complete correspondence with
the relation
v2(z, z ) = lim
n→∞
E
{
(�◦
γn(z))2
}
=
= lim
n→∞
1
4
E
{
(
◦
γn(z))2 + (
◦
γn(z))2 + 2
◦
γn(z)
◦
γn(z )
}
=
=
1
2
� (d2(z, z) + d2(z, z )) ,
where
◦
γn(z) is defined in (4.16) and in (4.22), and d2 is given by (4.2).
4.6. We will formulate now a general statement of similar nature. Its proof follows
the same strategy but is more tedious.
Theorem 4.3. Consider the GUE defined by (2.11) with β = 2, and denote γn(z) =
= Tr(M − z)−1, �z �= 0. Given integers p ≥ 0 and q ≥ 0 and η0 > 0, take a set of
points (z1, . . . , zp, zp+1, . . . , zp+q), such that |�zj | ≥ η0 > 0, j = 1, . . . , p + q, and
denote
◦
γR,n(zj) = �
[
γn(zj) − E{γn(zj)}
]
, j = 1, . . . , p,
◦
γI,n(zk) = �
[
γn(zk) − E{γn(zk)}
]
, k = p + 1, . . . , p + q.
Then the collection
Γ(n)
pq =
{◦
γR,n(zj), j = 1, . . . , p;
◦
γI,n(zk), k = p + 1, . . . , p + q
}
of p + q random variables converges in distribution as n → ∞ to the set of Gaussian
random variables
Γpq =
{◦
γR(zj), j = 1, . . . , p;
◦
γI(zk), k = p + 1, . . . , p + q
}
,
whose expectations are equal to zero and whose covariances are
Cov
{◦
γR(zj1),
◦
γR(zj2)
}
=
1
2
�
(
d2(zj1 , zj2) + d2(zj1 , zj2 )
)
, j1, j2 = 1, . . . , p,
Cov
{◦
γI(zk1),
◦
γI(zk2)
}
=
= −1
2
�
(
d2(zk1 , zk2) − d2(zk1 , zk2 )
)
, k1, k2 = p + 1, . . . , p + q,
Cov
{◦
γR(zj),
◦
γI(zk)
}
=
=
1
2
�
(
d2(zj , zk) − d2(zj , zk )
)
, j = 1, . . . , p, k = p + 1, . . . , p + q.
Remark 4.7. The above proof establishes the central limit theorem for the GUE and
we will give below a similar fact for the GOE. For a more general case see [5].
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808 L. A. PASTUR
Consider now analogous results for the GOE, corresponding to β = 1 in (2.11). Our
arguments will again be based on the differential formula (2.13), this time for the case of
real symmetric matrices, i.e., for β = 1.
It was explained above (see Remarks 3.2, 3.3) that the limiting form (3.30) of the
Normalized Counting Measure of eigenvalues as well as the order of magnitude of its
variance are the same for the GUE and the GOE (cf. (3.21) and (3.40)). The next theorem
is the central limit theorem for the GOE.
Theorem 4.4. Consider the GOE, defined in (2.11) with β = 1. Then for any z
such that �|z| ≥ η0 > 0 the random variable
◦
γR,n(z) := γR,n(z) − E{γR,n},
where γR,n(z) = �Tr(M − z)−1, converges in distribution to the Gaussian random
variable
◦
γR(z) whose expectation is zero and whose variance is
Var{◦
γR(z)} =
1
4
(
d1(z, z) + d1(z, z ) + 2d1(z, z )
)
,
where d1(z1, z2) is given by (4.13).
Proof. We will follow the scheme of the proof of Theorem 4.2, but using the differ-
entiation formula (2.13) for the case β = 1 of real symmetric matrices. We obtain the
following analog of the relation (4.23):
(z + 2w2fn(z))An(t) =
= −w2E
{
n−1 ◦
γ2
n(z) exp{it◦γR,n(z)}
}
+ w2E
{
n−1 ◦
γ2
n(z)
}
Fn(t)−
−itw2E
{[
n−1TrG3(z) + n−1TrG2(z)G(z)
]
exp{it◦γR,n(z)
}
−
− w2E
{[
n−1TrG2(z) − E{n−1TrG2(z)}
]
exp{it◦γR,n(z)}
}
.
The relation differs from (4.23) by the factor 1/2 in front of the second term of the r.h.s.
and by the last line (cf. the analogous term in ( 3.39)). In view of (4.6) the line can be
rewritten as
−w2E
{
d
dz
◦
gn(z) exp{it◦γR,n(z)}
}
.
Now by using an analogue of (4.25) for the derivative g′n(z), we can prove that this ex-
pression vanishes as n → ∞. The rest of the proof repeats literally that of Theorem 4.2.
We note that an analog of Theorem 4.3 is also valid for the GOE.
5. Other ensembles. We outline here analogs of Theorems 3.1, and 3.2 for certain
other ensembles, involving Gaussian random variables.
1. Ensembles with correlated Gaussian entries. We write again M = n−1/2W,
where now M and W are n × n, n = 2m + 1 real symmetric matrices and W =
= {Wjk}|j|,|k|≤m is the n×n central square of the double infinite real symmetric matrix
{Wjk}j,k∈Z, whose entries are Gaussian random variables such that
E{Wjk} = 0, E{Wj1k1Wj2k2} = Bj1−j2,k1−k2 + Bj1−k2,k1−j2 , (5.1)
where Bj,k satisfies the conditions
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 809
Bj,k = Bj,−k, Bj,k = Bk,j ,
∑
j,k∈Z
|Bj,k| := b < ∞. (5.2)
The case Bjk = w2δj0δk0 is the GOE.
Under these conditions the NCM of the corresponding ensemble converges weakly
with probability 1 to the limiting unit measure whose Stieltjes transform f is uniquely
defined by the relations:
f(z) =
1∫
0
e2πipf̂(z, p) dp, (5.3)
where f̂ is analytic in z, �f̂(z, p) · �z > 0, �z �= 0, |f̂(z, p)| ≤ |�z|−1 ∀p ∈ [0, 1)
and is uniquely determined by the equation, generalizing (3.32):
f̂(z, p) = −
z +
1∫
0
B̂(p, q)f̂(z, q)dq
−1
, (5.4)
where
B̂(p, q) =
∑
j,k∈Z
e2πip(j−k)Bj,k. (5.5)
To prove these facts we write as above, by using (2.17):
E{Gjk(z)} = −δjk
z
− 1
z
∑
|q|≤m
E
{
∆(n)
j−qGqk(z)
}
− E{T}, (5.6)
where
∆(n)
j−q =
1
n
∑
l,p∈[−m,m]
Bj−q,l−pGlp(z), (5.7)
and
T =
1
zn
∑
l,p,q∈[−m,m]
Bj−p,l−qGlp(z)Gqk(z).
We have by Schwarz inequality, (5.2), and by the inequality
∑
|l|∈m
|Glk(z)|2 ≤ |�z|−2 :
|T | ≤ 1
|�z|n
∑
|p|≤m
( ∑
l,q∈[−m,m]
|Bj−p,l−q| |Glp(z)|2×
×
∑
l,q∈[−m,m]
|Bj−p,l−q| |Gqk(z)|2
)1/2
≤
≤ 1
|�z|n
∑
|p|≤m
(∑
q∈Z
|Bj−p,q|
∑
|l|≤m
|Glp(z)|2
∑
l∈Z
|Bj−p,l|
∑
|q|≤m
|Gqk(z)|2
)1/2
≤
≤ 1
n|�z|3Bj−q,
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810 L. A. PASTUR
where
Bj =
∑
l∈Z
|Bj,l|.
Hence, the third term in the r.h.s. of (5.6) vanishes as n → ∞.
Let us show that the variance of (5.7) is of the order O(n−1). We use again the
Poincare – Nash inequality (2.20). This require the derivatives of ∆(n)
j−q with respect to
Mab. We have by (2.9)
∂∆(n)
j−q
∂Mab
= − 1
n
∑
l,p∈[−m,m]
Bj−q,l−pGla(z)Gbp(z),
and by an argument similar to that in the proof of (4.2) we obtain∣∣∣∣∣∂∆(n)
j−q
∂Mab
∣∣∣∣∣ ≤ Bj−q
n|�z|2 .
Thus, we have by (2.20) and (5.2):
Var
{
∆(n)
j−q
}
≤
≤
B2
j−q
n3|�z|4
∑
a1,b1,a2,b2∈[−m,m]
(
|Ba1−a2,b1−b2 | + |Ba1−b2,b1−a2 |
)
≤
2bB2
j−q
n|�z|4 .
This and (4.2) allow us to write (5.6) as
E{Gjk(z)} = −δjk
z
− 1
z
∑
|q|≤m
E
{
∆(n)
j−q
}
E{Gqk(z)} + O(n−1/2), |�z| �= 0.
By using the above relation, it is possible to obtain the following formulas for the limit f
of the expectation fn = E{n−1TrG(z)} :
f(z) = f0(z),
fj(z) = −1
z
− 1
z
∑
l∈Z
∆j−l(z)fl(z), ∆j(z) =
∑
l∈Z
Bj,l(z)fl(z), j ∈ Z.
Now, passing to the Fourier transforms in these formulas, we obtain (5.3) – (5.5). To prove
the convergence with probability 1 of gn(z) := n−1TrG(z) to the same limit, we use
again the Poincare – Nash inequality, leading to the O(n−2) bound of the variance of
gn(z) for |�z| �= 0. We have by (2.20), (5.1), (5.2), and (2.9):
Var{gn(z)} ≤
≤ n−3
∑
a1,b1,a2,b2∈[−m,m]
(
|Ba1−a2,b1−b2 | + |Ba1−b2,b1−a2 |
)
×
×E
{
|(G2)a1b1 | |(G2)a1b1 |(z)
}
.
Consider the contribution of the first term in the parentheses of the r.h.s. By using Schwarz
inequality for sums and for expectations, we obtain the bound
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 811
n−3
( ∑
a1,b1,a2,b2∈[−m,m]
|Ba1−a2,b1−b2 |E
{
|(G2)a1b1 |2
}
×
×
∑
a1,b1,a2,b2∈[−m,m]
|Ba1−a2,b1−b2 |E
{
|(G2)a1b1 |2
})1/2
≤
≤ b
n3
( ∑
a1,b1∈[−m,m]
E
{
|(G2)a1b1 |2
} ∑
a2,b2∈[−m,m]
E
{
|(G2)a1b1 |2
})1/2
≤
≤ b
n2|�z|4 .
The contribution of the second term of the r.h.s. of the above inequality for Var{gn(z)}
admits the same bound, implying an O(n−2) bound for Var{gn(z)}. This finishes an
outline of proof of the announced result for ensembles of random matrices with correlated
Gaussian entries. For earlier proofs see e.g. [10].
2. Deformed Wishart ensemble. The ensemble is defined by (3.2) in which now
M =
1
n
XX ′, (5.8)
where X = {Xjµ}n,pj=1,µ=1 is n× p matrix, X ′ is its transposed, and the entries of X
are i.i.d. Gaussian random variables of zero mean and of the variance x2 :
E{Xjµ} = 0, E
{
X2
jµ
}
= x2. (5.9)
Denoting again G(z) the resolvent of H, and gn(z) = n−1TrG(z), we find easily that
in this case
∂gn
∂Xjµ
= − 2
n2
n∑
k=1
(
G2(z)
)
jl
Xlµ, (5.10)
hence we have in view of Proposition 2.4 and (5.9):
Var{gn(z)} ≤ 4x2
n4
p∑
µ=1
E
n∑
l,m=1
(
G2(z)
)
jl
Xlµ(G2(z))jmXmµ
. (5.11)
Setting Xµ = {Xjµ}nj=1 ∈ R
n, we can write the r.h.s. of the inequality as
4x2
n4
p∑
µ=1
E
{
‖G2(z)Xµ‖2
}
.
By Schwarz inequality and (2.7) we have
‖G2(z)Xµ‖2 ≤ ‖G(z)‖4‖Xµ‖2 ≤ 1
|�z|4 ‖Xµ‖2 =
1
|�z|4
n∑
j=1
|Xjµ|2.
This, (5.9) and (5.11) imply
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812 L. A. PASTUR
Var{gn(z)} ≤ 4x4p
n3|�z|4 ≤ 8x4c
n2|�z|4 , (5.12)
if n is large enough, since p/n → c < ∞.
Note that the above bound is valid for any �z �= 0 and any “unperturbed” matrix
H(0) in (3.2). However, if H(0) is positively definite, then H is also positive defi-
nite and we have the bound O(1/n2) also for negative values of the spectral parameter:
z = −σ2. In this case we have to replace |�z| by σ2 in (5.12), because for any real
symmetric (Hermitian) matrix H we have ‖(H − z)−1‖ ≤
(
dist(spectrum H, z)
)−1
,
in particular, if H is positive definite, ‖(H + σ2)−1‖ ≤ σ−2. We will use this observa-
tion below.
It can be shown, by using (5.12) and an argument similar to that of the previous
section, that the Normalized Counting Measure of eigenvalues of ensemble (3.2), (5.8)
converge with probability 1 to the limiting distribution, whose Stieltjes transform solves
the equation (cf. (3.7)):
f(z) = f (0)
(
z − x2
1 + x2f(z)
)
, (5.13)
where f (0) is defined in (3.5). Analogous result is valid for complex valued Gaussian
matrices X and X∗ instead of X ′ in (5.8). This ensemble is called sometimes the
Laguerre ensemble because of use of Laguerre polynomials in the orthogonal polynomials
approach.
A bit more involved argument allows one to study a more general random matrix
than (5.9):
1
n
XTX ′,
where T is a p × p real symmetric matrix, whose NCM τp converges weakly to a
nonnegative measure τ. In this case we have instead of (5.13)
f(z) = f (0)
(
z −
∫
x2tτ(dt)
1 + x2tf(z)
)
.
Notice that the matrix T can also be random but independent of X. In this case we
have to assume that τ is nonrandom and that τp converges to τ with probability 1.
A particular case of this problem, corresponding to a diagonal T with i.i.d. diagonal
entries, was studied in [11] by another method.
3. Random matrices of the telecommunication theory. We consider the real symmetric
version, where the corresponding matrices have the form [12]:
B = X ′T ′(TXX ′T ′ + σ2)−1TX, (5.14)
in which X is as in (5.8), X ′ is its transposed, T is a n × n matrix such that the
Normalized Eigenvalue Measure of eigenvalues τn of TT ′ converges to a unit measure
τ. We outline an argument, showing that n−1TrB converges with probability one uni-
formly on compact sets of ]0,∞[ in σ to a nonrandom limit, given by formulas (5.24),
(5.25) below.
Assume first that the norms of TT ′ are uniformly bounded in n and note that
n−1TrB = 1 − σ2n−1Tr(A + σ2)−1, (5.15)
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 813
where
A = TXX ′T ′. (5.16)
Hence Var{n−1TrB} = Var
{
n−1Tr(A + σ2)−1
}
. Denoting as before gn(z) =
= n−1TrG(z), G(z) = (A− z)−1, we have for gn(−σ2) (cf. (5.10))
∂gn(−σ2)
∂Xjµ
= − 2
n2
(T ′G2(−σ2)TXµ)j .
Thus, by Proposition 2.4,
Var{gn(−σ2)} ≤ 8x4c
n2σ8
‖T‖4. (5.17)
To find the limit of E{gn(−σ2)} we can use the scheme similar to that of Section 3. We
will use however another scheme, outlined below and dated to [11] (see also [6, 13]).
For any two vectors X,Y of C
n denote by LX,Y the rank one matrix, acting on
Ψ ∈ C
n by the formula
LX,Y Ψ = (Ψ, X)Y. (5.18)
Then we can write (5.8) as M = n−1
∑p
µ=1
LXµ,Xµ
and A of (5.16) as
A =
1
n
p∑
µ=1
LYµ,Yµ
, Yµ = TXµ. (5.19)
Hence, we have by the resolvent formula
E{G(−σ2)} =
1
σ2
− 1
nσ2
p∑
µ=1
E{LYµ,GYµ
}. (5.20)
It is easy to show that for any Hermitian (real symmetric) matrix H, any Y ∈ C
n, and
any z that do not belong to the spectrum of H + LY,Y , we have
GY := (H + LY,Y − z)−1 = G− GLY.YG
1 + (GY, Y )
, G := (H − z)−1,
in particular
GY Y =
1
1 + (GY, Y )
GY. (5.21)
This formula and (5.20) yield
E{G(−σ2)} =
1
σ2
− 1
nσ2
p∑
µ=1
E
{
1
1 + n−1(GµYµ, Yµ)
LYµ,GµYµ
}
,
Gµ = G(−σ2)
∣∣
Yµ=0
.
(5.22)
Since the vectors {Xν}pν=1 are i.i.d., {Yν}pν=1 have the same property and since Gµ
does not contain Yµ, we have by (5.9) and (5.19)
Eµ{(GµYµ, Yµ)} = x2TrT ′GµT, Eµ{LYµ,GµYµ} = GµTT ′,
where Eµ{...} denotes the expectation with respect to Xµ, and if Varµ{...} is the
corresponding variance, then
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814 L. A. PASTUR
Varµ{n−1(GµYµ, Yµ)} ≤ Cx4
nσ4
‖T‖4,
where C is an absolute constant. This and the inequality (GµYµ, Yµ) ≥ 0 allow us
to replace n−1(GµYµ, Yµ) by n−1TrGµ in (5.22), and then, after applying once more
(5.21), by n−1TrT ′GT. We arrive to the relation
E{G(−σ2)} =
1
σ2
− x2
nσ2
p∑
µ=1
E
{
1
1 + x2n−1TrT ′GT
GTT ′
}
+ O(1/n).
Now, by using again Proposition 2.4, we prove an O(1/n2) bound for the variance of
n−1TrT ′GT (cf. (5.17)), which leads to the asymptotic formula
E{G(−σ2)} =
1 + x2hn
x2c
(
TT ′ +
σ2(1 + x2hn)
x2c
)−1
+ O(1/n), (5.23)
where
hn = E{n−1TrT ′G(−σ2)T}.
Applying the operation n−1Tr to (5.23) and to this relation, multiplied by T ′ from the
left and by T from the right, and by using an argument, similar to that in the proof of
Theorem 3.1, we prove that gn converges with probability 1 to a nonrandom limit f, hn
converges to h and f and h satisfy the following system of functional equations:
f =
(1 + x2h)
x2c
f0
(
−σ2(1 + x2h)
x2c
)
, (5.24)
h =
(1 + x2h)
x2c
f1
(
−σ2(1 + x2h)
x2c
)
,
where
f0(z) =
∞∫
0
τ(dλ)
λ− z
, f1(σ2) =
∞∫
0
λτ(dλ)
λ− z
, (5.25)
and τ is the limiting Normalized Counting Measure of eigenvalues of TT ′.
The above proof of (5.24) was given under the assumption that the norms of TT ′ are
uniformly bounded in n. The general case can be obtained by a standard truncation pro-
cedure, which is easy to carry out because the norm does not present in (5.24) and (5.25).
4. Wigner ensembles. A natural question is to which extent the above results, ob-
tained for ensembles with Gaussian variables, can be generalized. We will discuss shortly
this question for the Wigner ensembles, defined as follows (for technical convenience we
will consider in this subsection real symmetric matrices). Write the matrix M in the
form (3.3)
M = n−1/2W, (5.26)
where W =
{
W
(n)
jk
}n
j,k=1
with W
(n)
jk = W
(n)
kj ∈ R, 1 ≤ j ≤ k ≤ n. Suppose that
the random variables W
(n)
jk , 1 ≤ j ≤ k ≤ n are independent and that
E
{
W
(n)
jk
}
= 0, E
{
(W (n)
jk )2
}
= (1 + δjk)w2, (5.27)
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A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 815
i.e., the two first moments of the entries are as in the GOE case (see (3.36), (3.37)). In
other words, the probability law of the matrix W is
P(d1W ) =
∏
1≤j≤k≤n
F
(n)
j,k (dWjk), (5.28)
where for any 1 ≤ j ≤ k ≤ n F
(n)
j,k is a probability measure on the real line, satisfying
condition (5.27).
A sufficiently detailed study of the global regime of these ensembles is rather involved
(see e.g. [6, 7, 13, 14]). It is worth to note however that many of these results can be
obtained by applying a generalization of the method, used in previous sections and based
on resolvent identity and on differentiation formulas (2.13), and (2.17). The role of these
formulas in general case of the Wigner ensemble plays the following one [7].
Let ξ be a real valued and centered random variable, having p + 2 finite moments
for some positive integer p, and let Φ : R → C be a function, whose first (p + 1)
derivatives are bounded. Denote by κl, l = 1, 2, . . . , the cumulants (semiinvariants) of
ξ, i.e., the MacLaurin coefficients of logarithm of the characteristic function of ξ. Then
E{ξΦ(ξ)} =
p∑
l=1
κl+1
l!
E
{
Φ(l)(ξ)
}
+ εp, (5.29)
where
|εp| ≤ Cp sup
x∈R
∣∣∣Φ(p+1)(x)
∣∣∣E{
|ξ|p+2
}
, (5.30)
and Cp depends on p only. The cumulants can be expressed via the moments of ξ.
Namely, if µl = E{ξl}, and µ1 = 0, then
κ1 = 0, κ2 = µ2 = Var{ξ}, κ3 = µ3, κ4 = µ4 − 3µ2
2, (5.31)
etc. For the Gaussian random variable all cumulants but κ2 vanish, and the above formula
reduces to (2.15). Note that κ4 is called in statistics the excess of the random variable ξ.
It is an important ingredient of a simple statistical test to find that a given random variable
is not Gaussian.
We present now a result, whose proof is based on (5.29).
Theorem 5.1. Let n−1/2Ŵ be the GOE matrix (3.36), (3.37) and n−1/2W be the
Wigner matrix (5.26) – (5.28), satisfying the condition
sup
n
max
1≤j,k≤n
E
{
|W (n)
jk |3
}
:= w3 < ∞. (5.32)
Denote by G1(z) and G2(z) the resolvents of n−1/2Ŵ and n−1/2W. Then∣∣E{
n−1TrG1(z)
}
− E
{
n−1TrG2(z)
}∣∣ ≤ Cw3/n
1/2|�z|4, (5.33)
where C is an absolute constant.
Proof. Consider the “interpolating” random matrix (cf. (2.21))
M(t) =
√
t/n W +
√
(1 − t)/n Ŵ , 0 ≤ t ≤ 1, (5.34)
viewed as defined on the product of probability spaces of matrices W and Ŵ . In other
words, we assume that matrices W and Ŵ in (5.34) are independent. Denote again by
E{. . .} the corresponding expectation in the product space. It is evident that
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
816 L. A. PASTUR
M(1) = n−1/2W, M(0) = n−1/2Ŵ , (5.35)
and if G(z, t) is the resolvent of M(t), then we have by (5.35):
n−1TrG1(z) − n−1TrG2(z) =
1∫
0
d
dt
n−1TrG(z, t)dt
and by (2.9):
d
dt
n−1TrG(z, t) = n−1TrG2(z, t)
(
1
2
√
nt
W − 1
2
√
n(1 − t)
Ŵ
)
. (5.36)
Now we will apply the differentiation formula (5.29) with p = 1 to transform the first
term in parentheses. To this end we take into account that W and G2(z, t) are symmetric
and write the term as
1
2
√
nt
n∑
j=1
E
{
Wjj(G2(z, t))jj
}
+ 2
∑
1≤j<k≤n
E
{
Wjk(G2(z, t))kj
} .
Since {Wjk}1≤j<k≤n are independent, we can apply (5.29) to every term of the sums.
In view of (5.27) we obtain that the contribution of the first term in the parentheses of
(5.36) is
w2
2
√
n3t
n∑
j,k=1
(1 + δjk)E
{
∂
∂W
(n)
jk
(G2(z, t))kj
}
+ Rn, (5.37)
where
|R|n ≤ w3
2
√
n5t
n∑
j,k=1
sup
Mjk∈R
∣∣∣∣∣ ∂2
∂ (Mjk)
2 (G2(z, t))kj
∣∣∣∣∣ . (5.38)
It follows from the Gaussian differential formula (2.13) that the contribution of the second
term in the parentheses of (5.36) can be written in the same form as (5.37), but without
the remainder term Rn. By using formula (2.9), it is easy to show that the expressions
are
−w2
n2
E
{
TrG2TrG + TrG3
}
. (5.39)
Hence, the r.h.s. of (5.36) is equal to Rn. By using formula (2.9) twice, we find that the
second derivative in (5.38 ) is the sum of terms of the form n−1(G2)abGcdGef , where
a, b, . . . , f assume values j, k. Each of these terms is bounded by n−1|�z|−4 in view
of (2.7). We obtain that the remainder (5.38), hence the derivative (5.36), admits the
bound:
|R|n ≤ Cw3
n1/2 |�z|4
, (5.40)
where C is an absolute constant. This fact and the interpolating property (5.36)
yield (5.33).
Remarks. 5.1. A similar argument and (3.40) imply an O(n−1/2) bound for the
variance of n−1Tr(M−z)−1. This and a standard truncation procedure lead to the weak
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 6
A SIMPLE APPROACH TO THE GLOBAL REGIME OF GAUSSIAN ENSEMBLES . . . 817
convergence in probability of the NCM of a Wigner ensemble satisfying (5.27) and the
condition
lim
n→∞
n−2
∑
1≤j≤k≤n
∫
|W |≥τ√n
W 2F
(n)
j,k (dW ) = 0 ∀τ > 0. (5.41)
The condition is a matrix analog of the well known Lindeberg condition for the validity
of the central limit theorem. Hence the semicircle law is a common form of the limiting
eigenvalue counting measures for all Wigner ensembles, satisfying (5.27) and (5.41). For
these and other numerous results for the Wigner ensembles see e.g. [6, 7, 13, 14] and
references therein.
5.2. As another example of application of (5.29) we mention the asymptotic form of
the covariance of γn = Tr(M − z)−1 for the Wigner matrix (5.26) – (5.28), such that its
moment E
{∣∣W (n)
jk
∣∣5} is bounded uniformly in j, k and n [7]:
n2Cov
{
γn(z1)γn(z2)
}
= d1(z1, z2) + 2κ4h(z1)h(z2) + O(n−1/2),
where d1(z1, z2) is the covariance for the GOE, given by (4.13), h(z) = f2(z)(z2 −
−4w2)−1/2, and f is defined in (3.34).
Acknowledgements. I am grateful to Prof. P. Loubaton, whose questions motivated
the work. It was supported in part by the Fonds National de la Science (France) via the
ACI Program “Nouvelles Intergaces des Mathématiques”, project # NIM205.
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Received 14.02.2005
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|
| id | umjimathkievua-article-3645 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:46:23Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/05/e514a4246e4652e8873ccd0d009ea805.pdf |
| spelling | umjimathkievua-article-36452020-03-18T20:00:55Z A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices Простий підхід до глобального режиму гауссових ансамблів випадкових матриць Pastur, L. A. Пастур, Л. А. We present simple proofs of several basic facts of the global regime (the existence and the form of the non-random limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices, whose probability law involves the Gaussian distribution. The main difference with previous proofs is the systematic use of the Poincare - Nash inequality, allowing us to obtain the O(n - 2) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter. Наведено прості доведення низки основних фактів стосовно глобального режиму (існування та вигляд невипадкової граничної нормалізованої рахуючої міри для власних значень, центральна гранична теорема для сліду резольвенти) для ансамблів випадкових матриць, до ймовірнісного закону яких входить гауссів розподіл. Головна відмінність від попередніх доведень полягає у систематичному використанні нерівності Пуанкаре - Неша, що дозволило отримати оцінки порядку O(n - 2) для дисперсії нормалізованого сліду резольвенти, які справджуються до дійсної осі відносно спектрального параметра. Institute of Mathematics, NAS of Ukraine 2005-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3645 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 6 (2005); 790–817 Український математичний журнал; Том 57 № 6 (2005); 790–817 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3645/4020 https://umj.imath.kiev.ua/index.php/umj/article/view/3645/4021 Copyright (c) 2005 Pastur L. A. |
| spellingShingle | Pastur, L. A. Пастур, Л. А. A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices |
| title | A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices |
| title_alt | Простий підхід до глобального режиму гауссових ансамблів випадкових матриць |
| title_full | A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices |
| title_fullStr | A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices |
| title_full_unstemmed | A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices |
| title_short | A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices |
| title_sort | simple approach to the global regime of gaussian ensembles of random matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3645 |
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