Front. Math. China
DOI 10.1007/s11464-012-0259-5
Fluctuations of deformed
Wigner random matrices
Zhonggen SU
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
c Higher Education Press and Springer-Verlag Berlin Heidelberg 2012
Abstract Let Xn be a standard real symmetric (complex Hermitian) Wigner
matrix, y1 , y2 , . . . , yn a sequence of independent real random variables
independent of Xn . Consider the deformed Wigner matrix Hn,α = n−1/2 Xn +
n−α/2 diag(y1 , . . . , yn ), where 0 < α < 1. It is well known that the average
spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes
transform mn,α (z) converges in probability to the corresponding Stieltjes
transform m(z). In this paper, we shall give the asymptotic estimate for the
expectation Emn,α (z) and variance Var(mn,α (z)), and establish the central limit
theorem for linear statistics with sufficiently regular test function. A basic tool
in the study is Stein’s equation and its generalization which naturally leads to
a certain recursive equation.
Keywords Asymptotic expansion, deformed Wigner matrice, Gaussian
fluctuation, linear statistics, Stein’s equation
MSC 60F10, 60F15, 60G50
1 Introduction and main results
Assume that Xn = (xij ) is a standard n × n real symmetric (resp. complex
Hermitian) Wigner matrix, in which xii , 1 i n, are independent identically
distributed (i.i.d.) real random variables with mean zero and finite variance,
and xij , 1 i < j n, are i.i.d. real (resp. complex) random variables with
mean zero and variance 1. Assume that y1 , y2 , . . . , yn are i.i.d. real random
variables with mean zero and variance σ 2 > 0, and assume further that Xn and
yi ’s are independent of each other. Define
1
Xn
Hn,α = √ + α/2 diag(y1 , . . . , yn ),
n n
Received March 3, 2012; accepted November 13, 2012
E-mail: suzhonggen@zju.edu.cn
(1.1)
2
Zhonggen SU
where 0 < α < 1 and the subscript α in Hn,α emphasizes the dependence on
α. Hn,α is often referred to as the deformed Wigner matrix in the literature.
The purpose of this paper is to study the asymptotic fluctuations of eigenvalues
of Hn,α and to focus on the influence of the perturbation part (the diagonal
matrix) upon the spectral behaviors.
First, we note that the average spectral measure still follows the classical
Wigner semicircle law. Let λ1 , λ2 , . . . , λn be the n real eigenvalues, and define
1
1(λi x) ,
Fn,α (x) =
n
n
x ∈ R.
i=1
Then as n → ∞, we have
d
Fn,α (x) −→ ρ(x)
in probability,
(1.2)
where
1 4 − x2 , |x| 2,
2π
which is now referred to as the Wigner semicircular law (see [1,9] for details).
This can be equivalently described in terms of Stieltjes transform. For any
z with Im z = 0, let
∞
2
1
ρ(x)
dFn,α (x), m(z) =
dx.
mn,α(z) =
x
−
z
x
−z
−∞
−2
ρ(x) =
It is well known that (1.2) is equivalent to
P
mn,α (z) −→ m(z),
n → ∞.
(1.3)
Here, m(z) satisfies a simple equation
m(z) +
1
=0
z + m(z)
(1.4)
and is uniquely determined by the requirement Im(m(z)) > 0 for any z with
Im z > 0.
To see (1.3), we only need to note the independence of Xn and yi ’s and
to apply the general theorem for deformed Wigner matrices, in particular,
[7, Theorem 3.1]. Indeed, since y1 , y2 , . . . , yn are i.i.d. random variables, then
according to the law of large numbers, the limit of empirical measures from
the yi ’s concentrates only at the origin, whose associated Stieltjes transform is
−1/z.
Though the addition of diagonal matrix has no influence upon the global
limiting behavior of eigenvalues, the fluctuation of the largest eigenvalue around
the edge 2 obey no longer the Tracy-Widom law. This change was first observed
by Johansson [3]. Suppose that Xn is a Gaussian unitary ensemble (GUE),
Fluctuations of deformed Wigner random matrices
3
y1 , y2 , . . . , yn have finite moments up to order seven, and Hn,α is as in (1.1)
with α = 1/3. Then Johansson used the explicit expression for joint density of
eigenvalues and lengthy saddle point analysis to show
n1/6 (max λi − cn ) −→ ζ + ς,
d
i
n → ∞,
√
where cn ∼ 2 n is a centering constant, and ζ and ς are independent random
variables with the Tracy-Widom law and the normal law, respectively.
The perturbed matrix does not only change the limiting distribution of the
largest eigenvalue, but also deteriorates the precision of estimating m(z) by
2 the variance of entry H in H
mn,α (z). Denote by σij
ij
n,α . Trivially,
σii2 ∼ n−α ;
2
σij
= n−1 , i = j.
Therefore, the spread of the matrix is about nα instead of n. Contrary to the
fact that the variance of diagonal elements play no role in most of the studies
of standard Wigner matrices, the precision of estimating m(z) by mn,α (z) will
be mainly determined by n−α . Our main results read as follows.
Theorem 1.1 Assume that Xn is a Gaussian orthogonal ensemble (GOE),
y1 , y2 , . . . , yn are i.i.d. real random variables such that
Eyi = 0,
Var(yi ) = σ 2 > 0,
E|yi |q+2 < ∞,
where q is an integer such that qα > 2. Let Hn,α be as in (1.1). Then for any z
with Im z = 0, we have
Emn,α(z) = m(z) +
Theorem 1.2
we have
σ2
m3 (z)
·
+ O(n− min(3α,2)/2 ).
1 − m2 (z) nα
(1.5)
Under the hypotheses of Theorem 1.1, for any z with Im z = 0,
Var(mn,α (z)) =
|m2 (z)|2
σ2
·
+ O(n− min(2+3α,3+α)/2 ).
|1 − m2 (z)|2 n1+α
(1.6)
Having the estimates about expectation and variance, it is natural to ask
what the fluctuation distribution of mn,α (z) around its mean is. We shall show
that mn,α (z) − Emn,α(z) after properly normalized follows asymptotically the
normal distribution. More generally, the central limit theorem holds for linear
statistics of eigenvalues. To state the following theorem, we introduce the class
Hq of test functions:
∞
(1 + |t|)q |φ̂(t)|dt < ∞ ,
(1.7)
Hq = φ : R → R,
−∞
where
φ(λ) =
∞
−∞
eitλ φ̂(t)dt.
4
Zhonggen SU
Theorem 1.3 For every regular test function φ ∈ Hq , define
Nn,α(φ) =
n
φ(λi ).
i=1
Then under the assumption of Theorem 1.1, it follows
1
n(1−α)/2
d
(Nn,α (φ) − ENn,α(φ)) −→ N (0, Vφ ),
where Vφ is given by
∞ t 2
dt
ds
Vφ = σ
0
0
2
iλs
e
−2
ρ(λ)dλ
∞
−∞
2
−2
eiλt1 ρ(λ)dλt1 φ̂(t1 )dt1 .
(1.8)
(1.9)
Remark 1.1 Theorems 1.1–1.3 deal only with the case where Xn is a GOE
and yi ’s are arbitrary random variables. This is because we aim to look at the
influence of perturbation matrix. One can obtain similar asymptotic expansions
and central limit theorems for general Wigner matrix Xn . In particular, the
precision in (1.5) is still n−α , even for the case where Xn is a GUE. The cases
without any perturbation were already investigated at length in [4–7]. However,
the approximation precision for the GUE case is much stronger than for the
GOE case; the former is n−2 , while the latter is only n−1 .
Remark 1.2 The asymptotic expansions in (1.5) and (1.6) are valid for any
complex number z with non-zero imaginary part. This is easily seen when yi ’s
are i.i.d. normal random variables because one can apply the Poincaré-Nash
inequality to control the variance. The case of general yi will be obtained using
a recent work of Shcherbina [8] (see Lemma 2.3 below).
Remark 1.3 The case α = 1 was treated at length in [4,5]. We remark
that the coefficients in asymptotic expansions in (1.5) and (1.6), and the limit
variance Vφ in (1.9) all depend on the 4-th cumulant of entries off diagonal.
Instead, in the case 0 < α < 1, they depend only on σ 2 , the variance of yi .
The proofs of main theorems will be given in the next sections. The basic
idea is to use the Stein equation and its generalization as in a series of papers
such as [4–7]. A new ingredient is to obtain a good estimate from the induced
recursive equations.
2 Asymptotic expansion for expectation and variance
In this section, we shall give the proofs of Theorems 1.1 and 1.2. To simplify
notation, write
Hn,α = Wn + Yn ,
where
Xn
Wn = (Wij ) = √ ,
n
Yn = (Yij ) =
1
nα/2
diag(y1 , . . . , yn ).
Fluctuations of deformed Wigner random matrices
5
We begin by the following two basic lemmas, which can be found in [4] (see
(II.18), (III.33), and (III.41) therein).
Lemma 2.1 Let Hn = (Hij ) be a real symmetric matrix, define the Green
function G(z) = (Hn − z)−1 , and write G(z) = (Gij (z)). Then we have
(1)
1
1
1
,
|Gij (z)|2 ;
(2.1)
|Gij (z)| |Im z|
n
|Im z|2
i,j
(2) resolvent identity
Gij (z) = −
1
δij
+
Gik (z)Hkj ,
z
z
(2.2)
k
where δij is the Kronecker delta, and the sum is over k = 1, 2, . . . , n;
(3) differential formula
−Gpi (z)Gqi (z),
i = j,
∂Gpq (z)
=
∂Hij
−Gpi (z)Gqj (z) − Gpj (z)Gqi (z), i = j,
(2.3)
where we used the fact that Gij (z) = Gji (z) for any real symmetric matrix.
Lemma 2.2 (1) Stein equation. Assume that ξ is a normal variable with
mean zero and f is a differentiable function. Then
Eξf (ξ) = Var(ξ)Ef (ξ).
(2.4)
(2) Generalized Stein equation. Assume that ξ is a random variable with
finite (q + 2)-th moment and f is a differentiable function of order q + 1, where
q is a positive integer. Then
Eξf (ξ) =
q
κτ +1
τ =0
τ!
Ef (τ ) (ξ) + εq ,
(2.5)
where κτ +1 stands for the (τ + 1)-th cumulant of ξ and the remainder term εq
admits the bound
|εq | cq sup |f (q+1) (x)| E|ξ|q+2 ,
x∈R
cq 1 + (3 + 2q)q+2
.
(q + 1)!
We now prove Theorem 1.1. To this end, we shall need an upper bound for
variance of mn,α(z), which is essentially due to Shcherbina [8].
Lemma 2.3 Let
and denote
z = E + iη ∈ C,
η = 0,
G(z) = (Gij (z)) = (Hn,α − z)−1 .
6
Zhonggen SU
Then the following statements hold.
(1) For 0 δ < 1, we have
Var(mn,α (z)) C
1
1
+ 2 3+δ E|G11 (z)|1+δ ,
n1+α |η|3−δ
n |η|
(2.6)
where and in the sequel, C is a constant depending on neither η nor n and may
take different values from line to line. In particular,
C
Var(mn,α (z)) n1+α |η|4
(2)
E|mn,α(z) − Emn,α (z)|4 C
.
(2.7)
1
+
n2(1+α) η 8
1
n4 η 12
.
(2.8)
Proof The argument is very similar to that of [8, Proposition 2] with minor
modifications.
We first prove (1). Let Ek denote the conditional expectation with respect
to {Hij , 1 i k, i j n}. Then according to a standard martingale
argument, we have
Var(mn,α (z)) =
1
E
Gii (z) − E
Gii (z)
2
n
i
2
i
n
2
1 = 2
E Ek−1
Gii (z) − Ek
Gii (z) .
n
i
k=1
(2.9)
i
Denote by Ek the conditional expectation with respect to {Hkj , 1 j n}. By
the independence and the Cauchy-Schwarz inequality, the expectation of the
rightmost-hand side of (2.9) is further controlled by
E
Gii (z) − Ek
i
2
Gii (z) .
i
i
be the (n − 1) × (n − 1) matrix produced by deleting the i-th row
Let Hn,α
and column from Hn,α and denote by G(i) (z) the corresponding Green function.
Then
1
,
(2.10)
Gii (z) =
Hii − z − Zi
where
Zi = ai · G(i) ai ,
ai is the i-th column with Hii deleted and ‘·’ stands for usual scalar product.
For i = k, we have (see [2])
(k)
Gii (z) − Gii (z) =
(Gik (z))2
.
Gkk (z)
(2.11)
Fluctuations of deformed Wigner random matrices
7
It easily follows from the independence that
(k)
(k)
Gii (z) − Ek Gii (z) = Gii (z) − Gii (z) − Ek (Gii (z) − Gii (z)),
and thus, by (2.11), we have
E
Gii (z) − Ek
i
Gii (z)
2
i
1
1
2E
(Gik (z))2 − Ek
(Gik (z))2
Gkk (z)
Gkk (z)
i=k
2
i=k
2
+ 2E|Gkk (z) − Ek Gkk (z)| .
Define
A=−
1
,
Gkk (z)
B=
(Gik (z))2
.
(Gkk (z))2
i=k
Then we need to estimate
E
1 2
1
− Ek
,
A
A
E
B
B 2
− Ek
,
A
A
respectively. We shall focus on the latter, since the former is similar and simpler.
By (2.10), a simple algebra gives
Ek A = z + Ek ak · G(k) ak = z +
1
TrG(k)
n
(2.12)
and
Ek |A − Ek A|2 = Ek |Hkk + ak · G(k) ak − Ek ak · G(k) ak |2
σ2
2
+ α + Ek |ak · G(k) ak − Ek ak · G(k) ak |2
n n
σ2
2
1 2 (k)
(k)
= + α+ 2
|Gpq
(z)|2 + 2
|Gpp (z)|2
n n
n
n
=
p,q=k
p=k
σ2
2
2
+ α + 2 TrG(k) G(k) ,
n n
n
4 − 1 = 2.
where in the third equation, we used the fact EX12
Note also that
(Gki (z), i = k)(Hn(k) − z) = −Gkk (z)(Hki , i = k).
Then we immediately have
B = ak · (G(k) )2 ak ,
(2.13)
8
Zhonggen SU
and thus,
Ek |B − Ek B|2 = Ek |ak · (G(k) )2 ak − Ek ak · (G(k) )2 ak |2 It easily follows that
and
2
Tr(G(k) G(k) )2 .
n2
|Ek A|−1 Ek |Gkk (z)|,
1
1
TrG(k) G(k) 2 ,
n
|η|
(2.14)
|η|
1
TrG(k) G(k) z + TrG(k) .
n
n
(2.15)
Combining (2.12)–(2.14) together yields that for 0 δ < 1,
C n1α + n12 TrG(k) G(k)
Ek |A − Ek A|2
|Ek A|2
|Ek A|1−δ |Ek A|1+δ
1
TrG(k) G(k) /n2
C α 1−δ +
Ek |Gkk (z)|1+δ
n |η|
|z + n1 TrG(k) |1−δ
1
1
C α 1−δ +
Ek |Gkk (z)|1+δ .
n |η|
n|η|1+δ
Similarly, it holds that
1
1
Tr(G(k) G(k) )2 4 ,
n
|η|
|η|3
1
TrG(k) G(k) z + TrG(k) ,
n
n
(2.16)
and therefore,
CTr(G(k) G(k) )2
Ek |B − Ek B|2
|Ek A|2
n2 |Ek A|1−δ |Ek A|1+δ
CTr(G(k) G(k) )2
Ek |Gkk (z)|1+δ
n2 |z + n1 TrG(k) |1−δ
C
Ek |Gkk (z)|1+δ .
n|η|3+δ
Thus, we obtain
Ek
B
B
− Ek
A
A
2
B − Ek B 2
B A − Ek A 2
·
+ 4Ek
Ek A
A
Ek A
1
1
C α 3−δ +
Ek |Gkk (z)|1+δ .
n |η|
n|η|3+δ
4Ek
(2.17)
The same bound is valid for Ek | A1 − Ek A1 |2 . Hence, it follows from (2.9) that
Var(mn,α (z)) C
1
n1+α |η|3−δ
+
1
n2 |η|3+δ
E|G11 (z)|1+δ ,
(2.18)
Fluctuations of deformed Wigner random matrices
9
where we used the fact that Gkk (z) obeys a common distribution.
Letting δ = 0 in (2.18) and noting (2.1), we have
Var(mn,α (z)) C
n1+α |η|4
.
(2) A standard martingale argument again shows
4
1 E
G
(z)
−
E
Gii (z)
ii
k
3
n
i
i
k
8 1
8 B
1 4
B 4
− Ek
− Ek
3
E
+ 3
E
.
n
A
A
n
A
A
E|mn,α (z) − Emn,α(z)|4 k
k
We need to estimate
1 4
1
− Ek
,
A
A
E
E
B
B 4
− Ek
,
A
A
respectively. A simple algebra shows
E|Zk − Ek Zk |4 C
n2 η 4
,
which in turn implies
E|A − Ek A|4 = 23 E|Hkk |4 + 23 E|Zk − Ek Zk |4 C
1
1
+ 2 4 .
2α
n
n η
Also, it holds that
E|B − Ek B|4 = E|ak · (G(k) )2 ak − Ek ak · (G(k) )2 ak |4 C
n2 η 8
.
Thus, we easily obtain
E
B
B
− Ek
A
A
4
B − Ek B 4
B A − Ek A
·
27 E
+ 27 E
Ek A
A
Ek A
1
1
C 2α 8 + 2 12 .
n η
n η
4
A similar bound is also true for E| A1 − Ek A1 |4 . This completes the proof.
Remark 2.1 The martingale argument above provides an upper bound for
variance Var(mn,α (z)) for any z with Im z = 0. The explicit bound in (2.6) can
be used to estimate the variances of a class of linear eigenvalue statistics. This
was first observed by Shcherbina [8] for a general Wigner matrix. In the special
case of Gaussian random matrix, the Poincaré-Nash inequality can be used to
establish this bound, the reader is referred to [7] for more discussion.
10
Zhonggen SU
We now prove Theorem 1.1 provided that the yi ’s are normal random
variables.
Proposition 2.1 Let Xn be a GOE, and let g1 , g2 , . . . , gn are i.i.d. normal
random variables such that
Egi = 0,
Let
Var(gi ) = σ 2 > 0.
1
Xn
g
= √ + α/2 diag(g1 , . . . , gn ),
Hn,α
n n
(2.19)
where and in the sequel, the superscript g indicates that the diagonal matrix is
normal. Define
Ggn (z) =
1
g
Hn,α
−z
,
mgn,α (z) =
1
TrGgn (z).
n
Then for any z with Im z = 0, we have
Emgn,α (z) = m(z) +
Proof
σ2
m3 (z)
·
+ O(n− min(2α,1) ).
1 − m2 (z) nα
(2.20)
Write
g
g
= (Hij
),
Hn,α
Ggn (z) = (Ggij (z)),
Yng = (Yijg ) =
1
nα/2
diag(g1 , . . . , gn ).
Applying the resolvent identity (2.2), we get
1 1
g
EGgik (z)Hki
.
Emgn,α(z) = − +
z zn
(2.21)
i,k
g
is a normal matrix, the Stein equation (2.4) and differential formula
Since Hn,α
(2.3) immediately yield
EGgii (z)Hiig = −
2
n
+
σ2
E(Ggii (z))2
nα
(2.22)
and
1
g
= − E(Ggii (z)Ggkk (z) + (Ggik (z))2 ),
EGgik (z)Hki
n
Inserting (2.22) and (2.23) into (2.21), we have
i = k.
1 1
Emgn,α(z) = − − E(mgn,α (z))2
z z
g
σ2 1
− 1+α
E(Ggii (z))2 − 2 E
(Gik (z))2 .
zn
zn
i
i,k
(2.23)
(2.24)
Fluctuations of deformed Wigner random matrices
11
By virtue of the variance bound (2.7), it follows
E(mgn,α (z))2 = (Emgn,α (z))2 + O(n−(1+α) ).
Then we can rewrite (2.24) as
1 1
σ2 E(Ggii (z))2 + O(n−1 ),
Emgn,α(z) = − − (Emgn,α(z))2 − 1+α
z z
zn
(2.25)
i
where we used (2.1).
We next calculate i E(Ggii (z))2 . Applying the resolvent identity (2.2) again,
we get
1
1 1
E(Ggii (z))2 = − Emgn,α (z) +
E(Ggii (z))2 Yiig
n
z
zn
i
i
1
+
EGgii (z)Ggij (z)Wji .
zn
i,j
Similarly, it follows from the Stein equation (2.4) and (2.1) that
2σ 2 1
E(Ggii (z))2 Yiig = − 1+α
E(Ggii (z))3 = O(n−α )
n
n
i
i
and
1 1
EGgii (z)Ggij (z)Wji = − 2
E(3Ggii (z)(Ggij (z))2 + (Ggii (z))2 Ggjj (z))
n
n
i,j
i,j
1 g
(Gii (z))2 Ggjj (z) + O(n−1 ).
=− 2
n
i,j
Furthermore, note that
1 E(Ggii (z))2 Ggjj (z)
2
n
i,j
1 g
1
(Gii (z))2 +
E(Ggii (z))2 (mgn (z))0 ,
= Emgn (z) · E
n
n
i
i
where and in the sequel, ξ 0 stands for ξ − Eξ.
Apply the variance bound (2.7) to yield
Emgn,α(z)
1
g
2
+ O(n−α ).
E(Gii (z)) = −
n
z + Emgn,α(z)
(2.26)
i
Inserting (2.26) into (2.25) leads to
Emgn,α (z)
1 1
σ2
Emgn,α(z)
g
2
+O(n− min(2α,1) ). (2.27)
= − − (Emn,α(z)) + α ·
z z
zn z + Emgn,α (z)
12
Zhonggen SU
Observe
Emgn,α (z)
|η|−2 .
z + Emgn,α(z)
Then it follows
(Emgn (z))2 + zEmgn (z) + 1 + O(n−α ) = 0,
from which one readily derives
|Emgn,α (z) − m(z)| = O(n−α ).
(2.28)
In turn, substituting (2.28) into (2.27) implies
(Emgn,α(z))2 + zEmgn,α (z) + 1 −
σ2
m(z)
· α + O(n− min(2α,1) ) = 0.
z + m(z) n
Solving the above equation concludes the proof of (2.20).
Emgn,α (z),
we need a comparison
Having the asymptotic expansion for
relation to conclude the asymptotic expansion for Emn,α(z).
Lemma 2.4 Let Xn and yi ’s be as in Theorem 1.1, and let gi be independent
normal random variables with mean zero and variance σ 2 . Assume further that
g
as
all these random variables are mutually independent. Define Hn,α and Hn,α
in (1.1) and (2.19), respectively. Then for any z with Im z = 0, we have
Emgn,α (z) − Emn,α(z) = O(n−3α/2 ).
Proof
For 0 t 1, define
mii (t) =
√
t gi +
√
1 − t yi ,
1
Xn
Hn,α(t) = √ + α/2 diag(m11 (t), . . . , mnn (t)),
n n
and define accordingly,
G(z, t) =
1
,
Hn,α(t) − z
mn,α(z, t) =
1
TrG(z, t).
n
Then it obviously follows
mn,α (z, 0) = mn,α (z),
On the other hand, we have
mn,α (z, 1) − mn,α (z, 0)
1 1 ∂
TrG(z, t)dt
=
n 0 ∂t
mn,α (z, 1) = mgn,α(z).
(2.29)
Fluctuations of deformed Wigner random matrices
1
=−
2n
1
0
13
1
1
√ TrG2 (z, t)Yng − √
TrG2 (z, t)Yn dt.
1−t
t
(2.30)
Apply directly the Stein equation (2.4) to yield
1
1
ETrG2 (z, t)Yng =
E(G2 (z, t))ii Yiig
n
n
i
√
2σ 2 t = − 1+α
EG(z, t)ii G(z, t)2ij .
n
(2.31)
i,j
Similarly, by (2.5), we have
1
1
ETrG2 (z, t)Yn =
E(G2 (z, t))ii Yii
n
n
i
√
2σ 2 1 − t =−
EG(z, t)ii G(z, t)2ij + ε3 (t),
n1+α
(2.32)
i,j
where the remainder term
|ε3 (t)| c3 (1 − t)E|y1 |3 n−3α/2 .
Taking expectation in both sides of (2.30) and inserting (2.31) and (2.32) gives
Emgn,α(z) − Emn,α (z) = E(mn,α(z, 1) − mn,α (z, 0)) = O(n−3α/2 ).
This completes the proof.
Proof of Theorem 1.1
2.4.
It follows immediately from Proposition 2.1 and Lemma
Proof of Theorem 1.2
By the resolvent identity (2.2), we have
1 1
1
Gij (z)Wji .
Gii (z) = − + Gii (z)Yii +
z z
z
j
Hence, it is easy to see
1
Em0n,α (z)Gii (z)
n
i
1 Em0n,α (z)Gii (z)Yii
=
zn
i
1 +
Em0n,α(z)Gij (z)Wji
zn
Var(mn,α (z)) =
i,j
1
=: (IY + IW ),
z
(2.33)
14
Zhonggen SU
where and in the sequel,
m0n,α = mn,α − Emn,α.
We shall estimate IY and IW , respectively, below. For the sake of clarity,
we shall divide the lengthy calculation into several lemmas.
Lemma 2.5
IW = −2m(z)Var(mn,α (z)) + O(n− min(1+2α,(3+α)/2) ).
(2.34)
Proof Since the Wij are normal random variables, a direct application of the
Stein equation (2.4) leads to
1 E(Gij (z))2 m0n,α (z)
n2
i,j
1 EGii (z)Gjj (z)m0n,α (z)
− 2
n
i,j
2 EGil (z)Gjl (z)Gij (z).
− 3
n
IW = −
(2.35)
i,j,l
It easily follows from (2.7) and (2.1) that
1
1 E(Gij (z))2 m0n,α (z) E|m0n,α (z)| = O(n−(3+α)/2 ),
2
n
n|η|2
(2.36)
i,j
where and in the sequel, η = Im z.
Also, as estimated in [4] (see (III.41) therein), we get
2 EGil (z)Gjl (z)Gij (z) = O(n−2 ).
n3
(2.37)
i,j,l
Finally, we have
1 EGii (z)Gjj (z)m0n,α (z)
n2
i,j
= E(mn,α (z))2 m0n,α (z)
= 2Emn,α (z)Var(mn,α (z)) + E(m0n,α (z))2 m0n,α (z)
= 2Emn,α (z)Var(mn,α (z)) + O(n−3(1+α)/2 ),
(2.38)
where we used (2.8) to control the term E(m0n,α (z))2 m0n,α(z).
Plugging (2.36)–(2.38) into (2.35) yields
IW = −2Emn,α(z)Var(mn,α (z)) + O(n−(3+α)/2 ).
Now, (2.34) follows from (1.5) and (2.7). The proof is completed.
Fluctuations of deformed Wigner random matrices
15
To find the major contribution term of IY , we need the following upper
bound for Var(Gii (z)).
Lemma 2.6 For each 1 i n, we have
σ 2 |m(z)|2
+ O(n− min(3α,1+α)/2 ).
nα
Var(Gii (z)) =
Proof
(2.39)
Applying the resolvent identity (2.2) again, we get
Var(Gii (z)) = EG0ii (z)Gii (z) =
1
1
EG0ii (z)Gij (z)Wji ,
EG0ii (z)Gii (z)Yii +
z
z
j
where and in the sequel,
G0ii (z) = Gii (z) − EGii (z).
Use the generalized Stein equation (2.5) to yield
EG0ii (z)Gii (z)Yii
σ 2 ∂G0ii (z)Gii (z)
E
+ O(n−3α/2 )
nα
∂Yii
σ2
σ2
= − α E(Gii (z))2 G0ii (z) − α EGii (z)(Gii (z))2 + O(n−3α/2 ).
n
n
It is easy to see that
=
EG2ii (z)G0ii (z) = 2EGii (z)Var(Gii (z)) + E(G0ii (z))2 G0ii (z)
(2.40)
(2.41)
and
EGii (z)(Gii (z))2 = Emn,α(z)(Emn,α (z))2 + 2EGii (z)Var(Gii (z))
+ EGii (z)(G0ii (z))2 ,
(2.42)
where we used the fact that
EGii (z) = Emn,α(z).
Substituting (2.41) and (2.42) into (2.40), we have
EG0ii (z)Gii (z)Yii = −
σ2
Emn,α(z)(Emn,α (z))2
nα
+ O(n−α )Var(Gii (z)) + O(n−3α/2 ).
(2.43)
Similarly, using the generalized Stein equation (2.5) again, we have
j
EG0ii (z)Gij (z)Wji = −
1
2
E(Gij (z))2 G0ii (z) −
EGij (z)Gii (z)Gij (z)
n
n
j
j
16
Zhonggen SU
−
1
EGii (z)Gjj (z)G0ii (z).
n
(2.44)
j
It is easy to see that the first two terms are controlled by 3/(n|η|3 ), and the
third term is equal to
−Emn,α(z)Var(Gii (z)) − Em0n,α(z)Gii (z)G0ii (z).
(2.45)
Also, it follows from (2.7) that
|Em0n,α(z)Gii (z)G0ii (z)| = O(n−(1+α)/2 ),
which together with (2.44) and (2.45) in turn implies
EG0ii (z)Gij (z)Wji = −Emn,α(z)Var(Gii (z)) + O(n−(1+α)/2 ).
(2.46)
j
Combining (2.43) and (2.46) yields
Var(Gii (z)) = −
Emn,α(z)(Emn,α (z))2 σ 2
· α + O(n− min(3α,1+α)/2 ).
z + Emn,α(z)
n
Thus, (2.39) immediately holds from (1.5).
Lemma 2.7
|m(z)|2
1
+ O(n−α/2 ).
E(Gil (z))2 Gii (z) = −
n
z + 2m(z)
(2.47)
i,l
Proof
First, note that by (2.1) and (2.39),
1
1
E(Gil (z))2 G0ii (z) 2 (E|G011 (z)|2 )1/2 = O(n−α/2 ).
n
|η|
i,l
Then we need only to estimate
identity (2.2),
−
1
n
i,l
E(Gil (z))2 , which is by the resolvent
1 1 Emn,α(z)
+
E(Gil (z))2 Yll +
EGil (z)Gij (z)Wjl .
z
zn
zn
i,l
(2.48)
i,j,l
Applying the generalized Stein equation (2.5) and noting (2.1), we have
2σ 2 1
E(Gil (z))2 Yll = − 1+α
E(Gil (z))2 Gll (z) + O(n−3α/2 )
n
n
i,l
= O(n
−α
i,l
).
(2.49)
Fluctuations of deformed Wigner random matrices
17
It follows from the Stein equation (2.4) that
1
EGil (z)Gij (z)Wjl
n
i,j,l
2 3 =− 2
EGll (z)(Gij (z))2 − 2
EGil (z)Gjl (z)Gij (z).
n
n
i,j,l
(2.50)
i,j,l
Note that
1 E(Gil (z))2 Gjj (z)
n2
i,j,l
1
1
E(Gil (z))2 +
E(Gil (z))2 m0n,α (z)
n
n
i,l
i,l
1
2
E(Gil (z)) + O(n−(1+α)/2 )
= Emn,α(z)
n
= Emn (z)
(2.51)
i,l
and
1 EGil (z)Gjl (z)Gij (z) = O(n−1 ).
n2
(2.52)
i,j,l
Substituting (2.51) and (2.52) into (2.50) yields
1
1
EGil (z)Gij (z)Wjl = −Emn,α(z)
E(Gil (z))2 + O(n−(1+α)/2 ). (2.53)
n
n
i,j,l
i,l
Combining (2.48), (2.49), and (2.53) together immediately implies
Emn,α(z)
1
+ O(n−α ).
E(Gil (z))2 = −
n
z + 2Emn,α (z)
(2.54)
i,l
Now, (2.47) easily holds from (2.54) and (1.5).
We are now ready to give a precise estimate for IY .
Lemma 2.8
IY = −
Proof
σ2
|m(z)|2
· 1+α + O(n− min(2+3α,3+α)/2 ).
z + 2m(z) n
(2.55)
First, apply the generalized Stein equation (2.5) to yield
Em0n,α(z)Gii (z)Yii
=
q
∂ τ m0n,α(z)Gii (z)
κτ +1
E
+ εq ,
∂(Yii )τ
τ ! n(τ +1)α/2
τ =1
where κτ +1 is the (τ + 1)-th cumulant of y1 and the remainder term
εq cq (η)E|y1 |q+2 n−(q+2)α/2 .
(2.56)
18
Zhonggen SU
Fix τ 1 and use the Lebiniz differential formula for product of two functions:
τ τ −p
∂ τ m0n,α(z)Gii (z) Gii (z) ∂ p m0n,α(z)
τ ∂
=
·
.
(2.57)
∂(Yii )τ
∂(Yii )p
p ∂(Yii )τ −p
p=0
By (2.3), for any 1 p τ, we have
∂ τ −p Gii (z)
= (−1)τ −p (τ − p)! (Gii (z))τ −p+1
∂(Yii )τ −p
(2.58)
and
∂ p−1 (Gil (z))2
∂ p Gll (z)
=
−
= (−1)p p! (Gil (z))2 (Gii (z))p−1 .
∂(Yii )p
∂(Yii )p−1
(2.59)
Inserting (2.58) and (2.59) into (2.57), we have
τ
∂ τ m0n,α (z)Gii (z)
1 τ
=
(−1)
τ
!
(Gil (z))2 (Gii (z))τ −p+1 (Gii (z))p−1
∂(Yii )τ
n p=1
l
+ (−1)τ τ ! (Gii (z))τ +1 m0n,α (z).
For 1 p τ, by virtue of (2.1), we have
1
1
(Gil (z))2 (Gii (z))τ −p+1 (Gii (z))p−1 τ +2
n
|η|
(2.60)
(2.61)
l
where and in the sequel η = Im z.
Combining (2.56), (2.60), and (2.61) yields
σ2
1
1
Em0n,α(z)Gii (z)Yii = − 1+α ·
E(Gil (z))2 Gii (z)
n
n
n
i
i,l
+
(−1)τ κτ +1 1 ·
E(Gii (z))τ +1 m0n,α (z)
(τ +1)α/2
n
n
τ =1
i
q
+ O(n−(3α+2)/2 ) + O(n−(q+2)α/2 ).
(2.62)
Moreover, by (2.7), it holds that
|E(Gii (z))τ +1 m0n,α (z)| = O(n−(1+α)/2 ).
Therefore, the right-hand side of (2.62) is further written as
−
1
σ2
·
E(Gil (z))2 Gii (z)
n1+α n
i,l
+
q−2
(−1)τ κτ +1
τ =1
+ O(n
n(τ +1)α/2
−(3α+2)/2
·
1
E(Gii (z))τ +1 m0n,α (z)
n
i
) + O(n−(q+2)α/2 ).
(2.63)
Fluctuations of deformed Wigner random matrices
19
Lemma 2.7 has given a precise estimate for the sum over i, l in (2.63), and
thus, it remains to computing the terms with m0n,α (z). For τ = 1, 2, . . . , q − 2,
define
1
aτ +1 =
E(Gii (z))τ +1 m0n,α (z).
n
i
Next, we shall derive a recursive relation so as to provide a small upper bound
for aτ +1 . Applying the resolvent identity (2.2), we have
1
E(Gii (z))τ +1 m0n,α (z)Yii
zaτ +1 = − aτ +
n
i
1
+
E(Gii (z))τ Gij (z)m0n,α (z)Wji .
(2.64)
n
i,j
The Stein equation (2.4) easily gives
2τ + 1 1
E(Gii (z))τ Gij (z)m0n,α (z)Wji = −
E(Gii (z))τ (Gij (z))2 m0n,α (z)
n
n2
i,j
i,j
1 E(Gii (z))τ +1 Gjj (z)m0n,α (z)
− 2
n
i,j
2
E(Gii (z))τ Gij (z)Gil (z)Gjl (z).
− 3
n
i,j,l
Noting (2.7) and the trivial bound
1 (Gii (z))τ Gij (z)Gil (z)Gjl (z) |η|−(τ +3) ,
n
i,j,l
we have
1
E(Gii (z))τ Gij (z)m0n,α (z)Wji = O(n−(1+α) ).
n
(2.65)
i,j
Similarly, the generalized Stein equation (2.5) gives
1
E(Gii (z))τ +1 m0n,α(z)Yii
n
i
q−(τ +1)
=
τ1 =1
1 ∂ τ1 (Gii (z))τ +1 m0n,α (z)
κτ1 +1
·
E
∂(Yii )τ1
(τ1 )! n(τ1 +1)α/2 n i
+ O(n−(q−τ +1)α/2 ),
1 τ q − 2.
(2.66)
Similar to (2.60), we have
∂ τ1 (Gii (z))τ +1 m0n,α (z)
τ1 (τ + τ1 )!
(Gii (z))τ +τ1 +1 m0n,α (z)
=
(−1)
τ
∂(Yii ) 1
τ!
τ1
(−1)τ1 τ1 ! (τ + τ1 − p1 )!
(Gii (z))τ +τ1 −p1 +1
+
τ
!
(τ
−
p
)!
1
1
p =1
1
20
Zhonggen SU
1
(Gil (z))2 Gii (z)p1 −1 .
n
n
×
(2.67)
l=1
Substituting (2.67) into (2.66) and noting
n
1 (Gii (z))τ +τ1 −p1 +1 (Gil (z))2 Gii (z)p1 −1 = O(1),
n
(2.68)
i,l=1
we obtain
1
E(Gii (z))τ +1 m0n,α(z)Yii
n
i
q−(τ +1)
=
τ1 =1
(−1)τ1 (τ + τ1 )! κτ1 +1
aτ +τ1 +1 + O(n−(1+α) ).
τ1 ! τ ! n(τ1 +1)α/2
(2.69)
In turn, this with (2.65) implies the following recursive relation:
q−(τ +1)
zaτ +1 = −aτ +
τ1 =1
where
(−1)τ1 (τ + τ1 )! κτ1 +1
aτ +τ1 +1 + bτ +1 ,
τ1 ! τ ! n(τ1 +1)α/2
(2.70)
bτ +1 = O(n−(1+α) ).
We are now left to find a good estimate for each aτ from (2.70). Define
ζij =
(−1)j (i + j)! κj+1
,
i! j! n(j+1)α/2
⎛
and write
0 ζ11
−1 0
0 −1
..
..
.
.
0
0
⎛
⎜
⎜
⎜
Ξ=⎜
⎜
⎝
⎛
⎜
⎜
a=⎜
⎝
a2
a3
..
.
aq−1
1 i q − 2, 1 j q − (i + 1),
⎞
⎟
⎟
⎟,
⎠
⎜
⎜
ζ =⎜
⎝
⎞
ζ12 ζ13 · · · ζ1,q−3
ζ21 ζ22 · · · ζ2,q−4 ⎟
⎟
0 ζ31 · · · ζ3,q−5 ⎟
⎟,
⎟
..
..
..
⎠
.
.
.
0
0 ···
0
⎞
⎛
ζ1,q−2
b2 − a1
⎜
ζ2,q−3 ⎟
b3
⎟
⎜
⎟, b = ⎜
..
..
⎠
⎝
.
.
ζq−2,1
⎞
⎟
⎟
⎟.
⎠
bq−1
Then (2.69) becomes
za = Ξa + aq ζ + b.
(2.71)
Solving the equation system (2.70), we get
a = (zI − Ξ)−1 (aq ζ + b).
(2.72)
Fluctuations of deformed Wigner random matrices
Note that
a1 = O(n−(1+α) ),
21
aq = O(n−(1+α)/2 ),
and b is negligible. It is not hard to see
aτ = O(n−(q−τ +1)/2 ),
τ = 2, . . . , q − 1.
(2.73)
Substituting (2.73) and (2.47) into (2.63) concludes the assertion (2.55), as
desired.
To conclude the proof of Theorem 1.2, plug (2.34) and (2.55) into (2.33) to
yield
zVar(mn,α (z)) = − 2m(z)Var(mn,α (z)) −
σ2
|m(z)|2
· 1+α
z + 2m(z) n
+ O(n− min(2+3α,3+α)/2 ).
Now, we can easily obtain the asserted (1.6).
3 Gaussian fluctuation for linear statistics
In this section, we shall prove Theorem 1.3. We start with the following basic
lemma, which is an analog of Lemma 2.1 and could be found in [5].
Lemma 3.1 Assume that Hn = (Hij ) is a real symmetric matrix, and let
U (t) = eitHn ,
Then we have
(1) for s, t ∈ R,
|U (t)ij | 1,
t ∈ R.
U (s)jk U (t)kj = U (s + t)jj ;
(3.1)
k
(2) Duhamel identity:
U (t) = 1 + i
and in particular,
U (t)jl = δjl + i
t
0
U (s)Hn ds,
t
0
U (s)jk Hkl ds,
(3.2)
k
where δjl is the Kronecker delta;
(3) differential formula:
iUpj ∗ Uqj (t),
j = k,
∂U (t)pq
=
∂Hjk
k,
i(Upj ∗ Uqk (t) + Upk ∗ Uqj (t)), j =
(3.3)
22
Zhonggen SU
where ∗ denotes the convolution, for example,
t
Upj (s)Uqj (t − s)ds,
Upj ∗ Uqj (t) =
t > 0.
0
Moreover, assume that φ is a differentiable function. Then
φ (Hn )jj , j = k,
∂Tr(φ(Hn ))
=
∂Hjk
2φ (Hn )jk , j = k,
where
φ (Hn )jk = i
∞
−∞
(3.4)
U (t)jk tφ̂(t)dt.
Lemma 3.2 Let Hn and φ be as in Theorem 1.3, and let
U (t) = eitHn .
Then we have
(i) for each t,
and
Var(TrU (t)) C(1 + |t|)2q n1−α
(3.5)
Var(TrHn U (t)) C(1 + |t|)2q n1−α ;
(3.6)
(ii) for each 1 j n and each t,
Var(U (t)jj ) Cn−α ,
(3.7)
where C is a constant possibly depending on q.
Proof We start with the proof of (i), and only prove (3.5) since (3.6) is similar.
According to [8, Proposition 1], for every t, we have
∞
∞
2q
−η 2q−1
dηe η
Var(TrG(x + iη))dx. (3.8)
Var(TrU (t)) C(1 + |t|)
0
−∞
Hence, it suffices to estimate
∞
Var(TrG(x + iη))dx,
η > 0.
−∞
Note the following bound (see [8, Lemma 2]):
∞
E|G11 (x + iη)|1+δ dx Cη −δ ,
δ > 0.
−∞
Then it follows from (2.6) that
∞
n1−α
1
Var(TrG(x + iη))dx C
+ 3+2δ .
3
η
η
−∞
(3.9)
Fluctuations of deformed Wigner random matrices
23
Inserting (3.9) into (3.8) and taking δ so small that
3
δ<q− ,
2
we obtain (3.5), as desired.
Next, we turn to the proof of (ii). Write for clarity,
Xn
Wn := (Wij ) = √ ,
n
Hn = (Hij ),
For any t, t , let
Yn := (Yij ) =
1
diag(y1 , . . . , yn ).
nα/2
Pn (t , t) = EU (t )0jj U (t)jj ,
where and in the sequel,
U (t )0jj = U (t )jj − EU (t )jj .
Trivially,
Var(U (t)jj ) = Pn (t, t).
Since
Pn (−t , −t) = Pn (t , t),
we only focus on the case t , t > 0 below. Use the Duhamel formula (3.2) to
obtain
t
t
0
EU (t )jj U (s)jj Yjj ds + i
EU (t )0jj U (s)jk Wkj ds. (3.10)
Pn (t , t) = i
0
0
k
Applying the Stein equation (2.4) and the differential formula (3.3) to Wkj , we
have
2i t
EU (t )0jj U (s)jk Wkj =
EU (t1 )jk U (t − t1 )jj U (s)jk dt1
n
0
k
k
s
i
+
EU (t1 )jj U (s − t1 )kk U (t )0jj dt1
n
0
k
s
i
+
EU (t1 )jk U (s − t1 )jk U (t )0jj dt1 . (3.11)
n
0
k
By (3.1), (3.11) is further simplified as
k
EU (t )0jj U (s)jk Wkj =
2i
n
t
0
i
+
n
EU (t − t1 )jj U (s + t1 )jj dt1
0
s
ETrU (s − t1 )U (t1 )jj U (t )0jj dt1
24
Zhonggen SU
is
EU (s)jj U (t )0jj .
(3.12)
n
Similarly, applying the generalized Stein equation (2.5) and the differential
formula (3.3), we get
+
EU (t )0jj U (s)jj Yjj
σ 2 ∂U (t )0jj U (s)jj
E
+ iε3 (t , s)
nα
∂Yjj
iσ 2
= α [E(Ujj ∗ Ujj )(t )U (s)jj + E(Ujj ∗ Ujj )(s)U (t )0jj ] + iε3 (t , s),
n
where the remainder term
=
|ε3 (t , s)| (3.13)
c3 E|yj |3
∂ 2 U (t )jj U (s)jj
CE|yj |3
sup
(t + t )2 .
2
3α/2
3α/2
∂(Y
)
n
n
jj
Note a trivial relation
ETrU (s − t1 )U (t1 )jj U (t )0jj = ETrU (s − t1 )EU (t1 )jj U (t )0jj
+ EU (t1 )jj ETrU (s − t1 )U (t )0jj
+ E(TrU (s − t1 ))0 U (t1 )0jj U (t )0jj
and define
Rn (t , s) = −
−
−
−
−
σ2
[E(Ujj ∗ Ujj )(t )U (s)jj + E(Ujj ∗ Ujj )(s)U (t )0jj ]
nα
2 t
EU (t − t1 )jj U (s + t1 )jj dt1
n 0
1 s
EU (t1 )jj ETrU (s − t1 )U (t )0jj dt1
n 0
1 s
E(TrU (s − t1 ))0 U (t1 )0jj U (t )0jj dt1
n 0
s
EU (s)jj U (t )0jj − ε3 (t , s).
n
It follows from (3.5) that
|ETrU (s − t1 )U (t)0jj | C(1 + t)q n(1−α)/2
and
|E(TrU (s − t1 ))0 U (t1 )0jj U (t )0jj | C(1 + t)q n(1−α)/2 .
Combined, it is now easy to see for every t, t > 0,
|Rn (t , s)| C
((1 + t)q + (t + t )2 ),
nα
0 s t.
(3.14)
Fluctuations of deformed Wigner random matrices
25
Then, inserting (3.11) and (3.13) into (3.10), we obtain the equation
Pn (t , t) = −
t
ds
0
0
s
Qn (s − t1 )Pn (t , t1 )dt1 −
t
0
Rn (t , s)ds,
(3.15)
where Qn is defined by
1
ETrU (u).
(3.16)
n
Fix t for the moment and view Pn (t , t) as a function with respect to argument
t. Apply now [5, Proposition 2.1] to produce a unique solution
Qn (u) =
Pn (t , t) = −
t
0
Tn (t − s)Rn (t , s)ds,
(3.17)
Tn (t − s)Rn (t, s)ds.
(3.18)
n )−1 .
where Tn ↔ (z + Q
Letting t = t in (3.17), we have
Pn (t, t) = −
0
t
According to the semicircle law, it follows
2
1
eixu x2 − 4 dx,
Qn (u) → Q(u) :=
2π −2
(3.19)
and the convergence is uniform over any finite interval. In turn this implies
Tn (u) → T (u)
(3.20)
−1 . Note
uniformly over any finite interval, where T ↔ (z + Q)
z 1 2
z − 4,
Q(z)
= m(z) = − +
2 2
and therefore,
i
T (u) =
2π
L
eizu
dz
z + Q(z)
eizu
dz
L z + m(z)
i
m(z)eizu dz
=−
2π L
2
1
eixu x2 − 4 dx.
=−
2π −2
=
i
2π
(3.21)
Thus, |Tn (ζ)| 2 on any finite interval of R whenever n is sufficiently large
since |T (ζ)| 1.
26
Zhonggen SU
Proof of Theorem 1.3
The proof is basically along the line of [5]. Write
Nn0 =
1
(Nn (φ) − ENn (φ)).
n(1−α)/2
We shall use the classical characteristic function method, namely, to show for
each x ∈ R,
0
2
2
Zn (x) := EeixNn → e−σ Vφ x /2 , n → ∞.
(3.22)
This in turn will be established by a subsequence technique. In particular, we
shall show that if Zn (x) → Z(x), then
Zn (x) → −σ 2 Vφ xZ(x).
Let
0
en (x) = eixNn ,
(3.23)
e0n (x) = en (x) − Een (x).
Note that Nn (φ) can be written as
Nn (φ) =
∞
TrU (t)φ̂(t)dt.
−∞
Then it follows that
Zn (x)
=
i
n(1−α)/2
Define
Υn (t, x) =
−∞
ETrU (t)e0n (x)φ̂(t)dt.
1
n(1−α)/2
so that
Zn (x)
∞
∞
=i
−∞
ETrU (t)e0n (x),
Υn (t, x)φ̂(t)dt.
By (3.5), the Schwarz inequality, and |en (x)| 1, we have
|Υn (t, x)| 1
n(1−α)/2
(Var(TrU (t)))1/2 C(1 + |t|)q .
Similarly, it follows that
1
∂Υn (t, x)
= (1−α)/2 |ETrHn U (t)e0n (x)| C(1 + |t|)q .
∂t
n
Still, in view of φ ∈ Hq , we have
|x|
∂Υn (t, x)
= 1−α |E(TrU (t))0 (Nn (φ))0 en (x)|
∂x
n
∞
|x|
E|(TrU (t))0 | |(TrU (t1 ))0 | |φ̂(t1 )|dt1
1−α
n
−∞
C|x|(1 + |t|)q ,
(3.24)
Fluctuations of deformed Wigner random matrices
27
where and in the sequel,
(TrU (t))0 = TrU (t) − ETrU (t),
(Nn (φ))0 = Nn (φ) − ENn (φ).
Therefore, we conclude that the sequence Υn is bounded and equi-continuous
on any finite set of R2 . It reduces to proving that any uniformly converging
subsequence of Υn has the same limit.
Now, using the Duhamel formula (3.2), we have
t
i
EU (s)jj e0n (x)Yjj ds
Υn (t, x) = (1−α)/2
n
0
j
t
i
EU (s)jk e0n (x)Wkj ds
+ (1−α)/2
n
0
j,k
=: Υn,1 (t, x) + Υn,2 (t, x).
(3.25)
Since the Wkj are normal variables, as discussed in [5, Section 2.2], one easily
proves
Υn,2(t, x) → 0.
Thus, we need only to focus on the term Υn,1 (t, x) below. Apply the generalized
Stein equation (2.5), and the differential formulae (3.3) and (3.4) to obtain
EU (s)jj e0n (x)Yjj =
q
∂ τ U (s)jj e0n (x)
κτ +1
E
+ εq,j (s)
τ
(τ +1)α/2
∂(Y
)
τ
!
n
jj
τ =1
q
∂en (x) κτ +1
σ2
+
= α EU (s)jj
n
∂Yjj
τ ! n(τ +1)α/2
τ =2
τ ∂ τ −m U (s)jj ∂ m en (x)
τ
×
·
E
∂(Yjj )τ −m ∂(Yjj )m
m
m=1
+
q
∂ τ U (s)jj 0
κτ +1
E
en (x) + εq,j (s),
τ
(τ +1)α/2
∂(Y
)
τ
!
n
jj
τ =1
(3.26)
where the remainder term
|εq,j (s)| cq E|y1 |q
∂ q+1 U (s)jj e0n (x)
sup
.
∂(Yjj )q+1
n(q+2)α/2
(3.27)
By virtue of (3.24) and (3.25), (3.23) will immediately hold if we prove the
following limits holds:
(i)
t
∂en (x)
1
EU (s)jj
ds
∂Yjj
n(1+α)/2 0 j
∞ 2
t 2
iλs
iλt1
ds
e ρ(λ)dλ
e ρ(λ)dλ t1 φ̂(t1 )dt1 ;
→ xZ(x)
0
−2
−∞
−2
28
Zhonggen SU
(ii) for each 2 τ q and 1 m τ,
1
n(τ α+1)/2
t
∂ τ −m U (s)jj ∂ m en (x)
E
·
ds → 0;
∂(Yjj )τ −m ∂(Yjj )m
0
j
(iii) for each 1 τ q,
1
n(τ α+1)/2
t
∂ τ U (s)jj 0
E
e (x)ds → 0.
∂(Yjj )τ n
0
j
First, we prove (i). Note by (3.4) that
EU (s)jj
ix
∂en (x)
= (1−α)/2 EU (s)jj φ (Hn )jj en (x)
∂Yjj
n
∞
x
EU (s)jj U (t1 )jj en (x)t1 φ̂(t1 )dt1
= − (1−α)/2
n
−∞
and a trivial equation
EU (s)jj U (t1 )jj en (x)
= EU (s)jj E(U (t1 )jj )0 e0n (x) + EU (t1 )jj E(U (s)jj )0 e0n (x)
+ Een (x)E(U (s)jj )0 (U (t1 )jj )0 + E(U (s)jj )0 (U (t1 )jj )0 e0n (x)
+ EU (t1 )jj EU (s)jj Een (x).
According to (3.7) and the control convergence theorem, we have
t
1
lim
n→∞ n(1+α)/2
0
t
EU (s)jj
j
∞
∂en (x)
ds
∂Yjj
lim EU (t1 )11 EU (s)11 Een (x)t1 φ̂(t1 )dt1
∞ 2
iλs
iλt1
= − xZ(x)
ds
e ρ(λ)dλ
e ρ(λ)dλ t1 φ̂(t1 )dt1 ,
=−x
ds
−∞ n→∞
2
t
0
0
−2
−∞
−2
where in the last equation, we used the assumption Een (x) → Z(x) and the
fact that for any t,
1
=
EU (t)ll →
n
n
EU (t)11
l=1
2
eiλt ρ(λ)dλ.
−2
Turn to the proof of (ii). Note that when m 1,
x
∂ m en (x)
= − (1−α)/2
∂(Yjj )m
n
∞
−∞
∂ m−1 U (t1 )jj en (x)
t1 φ̂(t1 )dt1 .
∂(Yjj )m−1
Fluctuations of deformed Wigner random matrices
29
Therefore, it suffices to prove
t ∞ 1
∂ τ −m U (s)jj ∂ m−1 U (t1 )jj en (x)
1
ds
E
·
t1 φ̂(t1 )dt1 → 0.
∂(Yjj )τ −m
∂(Yjj )m−1
n(τ −1)α/2 0
−∞ n j
Since τ 2, we only need to prove that the double integral is finite. In view of
(3.1) and (3.3), for any t, l 0, we have
∂ l U (t)jj
tl .
∂(Yjj )l
Also, for φ ∈ Hq , we have
∂U (t1 )jj
x
∂U (t1 )jj en (x)
=
en (x)− (1−α)/2 en (x)
∂Yjj
∂Yjj
n
∞
−∞
U (t1 )jj U (t2 )jj t2 φ̂(t2 )dt2 ,
which implies
∂U (t1 )jj en (x)
|x|
t1 + (1−α)/2
∂Yjj
n
∞
−∞
|t2 | |φ̂(t2 )|dt2 Cφ (x)(1 + t1 ),
where Cφ (x) is a constant possibly depending on x and φ.
Similarly, taking repeatedly derivatives yields
∂ l U (t1 )jj en (x)
Cφ (x)(1 + t1 )l .
∂(Yjj )l
Combined, we get
∂ τ −m U (s)jj ∂ m−1 U (t1 )jj en (x)
·
Cφ (x)sτ −m (1 + t1 )m−1 .
∂(Yjj )τ −m
∂(Yjj )m−1
Thus, we have proved that the above double integral with respect to s and t1
is finite, and therefore, the limit is 0, as desired.
Finally, we prove (iii). To this end, for τ 1 and t 0, define
Δτ =
sup
0t1 ,...,tτ t
τ
E
U (tl )jj e0n (x) .
j
l=1
In view of (3.3), it is easy to see
t
∂ τ U (s)jj 0
tτ +1
Δτ +1 .
E
e
(x)ds
n
∂(Yjj )τ
τ +1
0
j
It suffices to prove for each 1 τ q,
Δτ +1
(τ
n α+1)/2
→ 0.
(3.28)
30
Zhonggen SU
It follows from the Duhamel formula (3.1) that for each m 2,
E
m
U (tl )jj e0n (x) = E
m−1
l=1
U (tl )jj e0n (x) + i
l=1
tm
+i
0
E
m−1
k
0
tm
E
m−1
U (tl )jj U (u)jj e0n (x)Yjj du
l=1
U (tl )jj U (u)jk e0n (x)Wkj du.
(3.29)
l=1
Write
γ1,j (u) = iE
m−1
U (tl )jj U (u)jj e0n (x)Yjj
(3.30)
l=1
and
γ2,j (u) = i
E
k
m−1
U (tl )jj U (u)jk e0n (x)Wkj .
(3.31)
l=1
The Stein equation yields
γ2,j (u) = −
m−1
m−1
2 E(U jj ∗ U jk )(tr )
U (tl )jj U (u)jk e0n (x)
n r=1
l=r
k
−
1
E
n
m−1
k
−
−
U (tl )jj (U jk ∗ U jk )(u)e0n (x)
l=1
2σ 2 x n(3−α)/2
1
E
n
k
k
m−1
E
m−1
U (tl )jj U (u)jk φ (Hn )jk en (x)
l=1
U (tl )jj (U jj ∗ U kk )(u)e0n (x)
l=1
=: − δ1,j (u) − δ2,j (u) − δ3,j (u) − δ4,j (u).
(3.32)
And δ4,j can be further written as
δ4,j (u) =
u
0
m−1
U (tl )jj Ujj (v)
l=1
u
=
0
E
Qn (u − v)E
m−1
1
TrU (u − v)e0n (x)dv
n
U (tl )jj Ujj (v)e0n (x)dv
l=1
u
+
0
m−1
1
E(TrU (u − v))0
U (tl )jj Ujj (v)e0n (x)dv,
n
(3.33)
l=1
where Qn is as in (3.16).
Put
u
m−1
1
0
E(TrU (u − v))
U (tl )jj Ujj (v)e0n (x)dv.
γ3,j (u) =
0 n
l=1
(3.34)
Fluctuations of deformed Wigner random matrices
31
Fix t1 , t2 , . . . , tm−1 , and define
Pn,j (v) = E
m−1
U (tl )jj U (v)jj e0n (x)
l=1
−E
m−1
U (tl )jj e0n (x).
(3.35)
l=1
Substituting (3.30)–(3.34) into (3.29), we obtain the equation
tm
u
tm
du
Qn (u − v)Pn,j (v)dv −
Rn,j (u)du,
Pn,j (tm ) = −
0
0
0
where
Rn,j (u) = E
m−1
U (tl )jj e0n (x)
l=1
u
0
Qn (v)dv
+ γ1,j (u) + δ1,j (u) + δ2,j (u) + δ3,j (u) + γ3,j (u).
According to [5, Proposition 2.1], the equation has a unique solution
tm
Tn (tm − u)Rn,j (u)du,
Pn,j (tm ) =
(3.36)
(3.37)
0
n )−1 .
where Tn ↔ (z + Q
It now follows directly from (3.35)–(3.37) that
E
m
U (tl )jj e0n (x)
l=1
m−1
duTn (tm − u)
Qn (v)dv E
U (tl )jj e0n (x)
= 1+
0
0
l=1
tm
Tn (tm − u)(γ1,j (u) + δ1,j (u) + δ2,j (u) + δ3,j (u) + γ3,j (u))du. (3.38)
+
tm
u
0
Observe that Qn (ζ) converge uniformly on any finite interval of R to
2
1
eiζλ 4 − λ2 dλ,
Q(ζ) =
2π −2
(3.39)
n (z) converge uniformly on the L = (−∞ − iε, ∞ − iε) with
and therefore, Q
ε > 0 to m(z). This in turn indicates that Tn → T uniformly on any finite
interval of R. Thus, |Tn (ζ)| 2 on any finite interval of R whenever n is
sufficiently large since |T (ζ)| 1.
Applying the generalized Stein equation, we have
γ1,j (u)
q
=i
τ1
∂ τ1
κτ1 +1
E
τ ! n(τ1 +1)α/2
=1 1
m−1
l=1
U (tl )jj U (u)jj e0n (x)
+ iεq (u)
∂(Yjj )τ1
32
Zhonggen SU
m−1
τ1 U (tl )jj U (u)jj ∂ m1 en (x)
∂ τ1 −m1 l=1
κτ1 +1
τ1
=i
·
E
τ1 −m1
(τ1 +1)α/2
m
∂(Y
)
∂(Yjj )m1
τ
!
n
1
jj
1
τ1 =1
m1 =1
m−1
q
∂ τ1 l=1
U (tl )jj U (u)jj 0
κτ1 +1
+i
E
en (x) + iεq (u),
(τ
+1)α/2
1
∂(Yjj )τ1
τ1 ! n
q
τ1 =1
where
|εq (u)| Cn−(q+2)α/2 .
(3.40)
For each 1 m1 τ1 , we have
m−1
U (tl )jj U (u)jj
∂ τ1 −m1 l=1
m · · · (m + τ1 − m1 − 1)tτ1 −m1
∂(Yjj )τ1 −m1
and
∂ m1 en (x)
Cφ,x n−(1−α)/2 .
∂(Yjj )m1
(3.41)
(3.42)
Also, for each τ1 1, we have
∂ τ1 m−1 U (tl )jj U (u)jj
l=1
E
e0n (x) m · · · (m + τ1 − 1)tτ1 Δm+τ1 .
∂(Yjj )τ1
j
It follows from (3.1) and |en (x)| 1 that for each 1 j n,
E(U jj ∗ U jk )(tr )
m−1
U (tl )jj U (u)jk e0n (x) 2t,
(3.43)
l=r
k
k
E
m−1
U (tl )jj (U jk ∗ U jk )(u)e0n (x) 2t,
(3.44)
l=1
and
k
E
m−1
U (tl )jj U (u)jk φ (Hn )jk en (x) l=1
∞
−∞
|t| |φ̂(t)|dt.
(3.45)
2t
,
n
(3.46)
These together with (3.32) immediately imply
|δ1,j (u)| 4(m − 1)t
,
n
and
|δ3,j (u)| 2σ 2 x
n(3−α)/2
|δ2,j (u)| ∞
−∞
|t| |φ̂(t)|dt.
(3.47)
On the other hand, by (3.7), we have
|γ3,j (u)| Cn−(1+α)/2 .
(3.48)
Fluctuations of deformed Wigner random matrices
33
Now, taking sum over j in (3.38) and noting the preceding bounds, we derive
Δm
q
C Δm−1 +
τ1 =1
1
n(τ1 +1)α/2
Δm+τ1 + n
(1−α)/2
,
m 2,
(3.49)
where C is a positive constant not depending on n.
According to (3.5),
Δ1 = O(n(1−α)/2 ).
Thus, solving (3.49), we obtain
Δm = O(n(1−α)/2 ),
m 2,
from which (3.28) holds. This completes the proof of (iii), and therefore,
concludes the assertion.
Acknowledgements Much of the work was done when the author was visiting the
Department of Mathematics, Harvard University, under the project from the Y. C. Tang
Disciplinary Development Fund, Zhejiang University. The author thanks Professor H. T. Yau
and Professor S. T. Yau for their hospitality during the visit. The referees’ careful reading
helps to improve the presentation of the paper. This work was Partially supported by the
National Natural Science Foundation of China (Grant No. 11071213), the Natural Science
Foundation of Zhejiang Province (No. R6090034), and the Doctoral Program Fund of
Ministry of Education (No. J20110031).
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