On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups
We consider first the n x n random matrices Hn = An +Un*BnUn, where An and Bn are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as n →∞, and Un is unitary uniformly distributed over U(n). We find the leading term of asymptotic expansion for the covariance of element...
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irk-123456789-1068092016-10-06T03:02:31Z On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups Vasilchuk, V. We consider first the n x n random matrices Hn = An +Un*BnUn, where An and Bn are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as n →∞, and Un is unitary uniformly distributed over U(n). We find the leading term of asymptotic expansion for the covariance of elements of resolvent of Hn and establish the Central Limit Theorem for the elements of suffciently smooth test functions of the corresponding linear statistics. We consider then analogous problems for the matrices Wn = SnUn*TnUn, where Un is as above and Sn and Tn are non-random unitary matrices having limiting NCM's as n →∞. Рассмотрены сначала n x n случайные матрицы вида Hn = An +Un*BnUn, где An и Bn - эрмитовы, имеющие предельную нормированную считающую меру (НСМ) собственных значений при n →∞, и Un - унитарные, распределенные равномерно по U(n). Найден ведущий член асимптотического разложения ковариации элементов резольвенты Hn и доказана Центральная Предельная Теорема для элементов достаточно гладких тестовых функций соответствующих линейных статистик. Затем аналогичные задачи рассмотрены для матриц вида Wn = SnUn*TnUn, где Un такая же, а Sn и Tn - неслучайные унитарные матрицы, имеющие предельные НСМ n →∞. 2014 Article On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups / V. Vasilchuk // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 451-484. — Бібліогр.: 9 назв. — англ. 1812-9471 DOI: http://dx.doi.org/10.15407/mag10.04.451 http://dspace.nbuv.gov.ua/handle/123456789/106809 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We consider first the n x n random matrices Hn = An +Un*BnUn, where An and Bn are Hermitian, having the limiting normalized counting measure (NCM) of eigenvalues as n →∞, and Un is unitary uniformly distributed over U(n). We find the leading term of asymptotic expansion for the covariance of elements of resolvent of Hn and establish the Central Limit Theorem for the elements of suffciently smooth test functions of the corresponding linear statistics. We consider then analogous problems for the matrices Wn = SnUn*TnUn, where Un is as above and Sn and Tn are non-random unitary matrices having limiting NCM's as n →∞. |
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Vasilchuk, V. |
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Vasilchuk, V. On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups Журнал математической физики, анализа, геометрии |
| author_facet |
Vasilchuk, V. |
| author_sort |
Vasilchuk, V. |
| title |
On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_short |
On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_full |
On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_fullStr |
On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_full_unstemmed |
On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups |
| title_sort |
on the fluctuations of entries of matrices whose randomness is due to classical groups |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/106809 |
| citation_txt |
On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups / V. Vasilchuk // Журнал математической физики, анализа, геометрии. — 2014. — Т. 10, № 4. — С. 451-484. — Бібліогр.: 9 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT vasilchukv onthefluctuationsofentriesofmatriceswhoserandomnessisduetoclassicalgroups |
| first_indexed |
2025-07-07T19:04:00Z |
| last_indexed |
2025-07-07T19:04:00Z |
| _version_ |
1837016073905373184 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 4, pp. 451–484
On the Fluctuations of Entries of Matrices whose
Randomness is due to Classical Groups
V. Vasilchuk
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail: vasilchuk@ilt.kharkov.ua
Received December 20, 2013, revised September 9, 2014
We consider first the n×n random matrices Hn = An +U∗
nBnUn, where
An and Bn are Hermitian, having the limiting normalized counting measure
(NCM) of eigenvalues as n → ∞, and Un is unitary uniformly distributed
over U(n). We find the leading term of asymptotic expansion for the co-
variance of elements of resolvent of Hn and establish the Central Limit
Theorem for the elements of sufficiently smooth test functions of the cor-
responding linear statistics. We consider then analogous problems for the
matrices Wn = SnU∗
nTnUn, where Un is as above and Sn and Tn are non-
random unitary matrices having limiting NCM’s as n →∞.
Key words: Random matrices, Central Limit Theorem, Limit Laws.
Mathematics Subject Classification 2010: 60F05, 15B52 (primary); 15A18
(secondary).
1. Introduction
In the recent decades the asymptotic properties of eigenvalues of various ran-
dom matrices have been of great interest (see, e.g., [3]). This paper deals with
the entries of functions of two classes of random matrices.
The first class consists of the n× n Hermitian random matrices
H = A + U∗BU (1.1)
and
H̃ = V ∗AV + U∗BU. (1.2)
Note that here and below we do not write the subindex n in various matrices.
c© V. Vasilchuk, 2014
V. Vasilchuk
We assume that A and B are non-random Hermitian matrices, V and U
are independent unitary random matrices uniformly distributed over the unitary
group U(n), i.e., having the normalized Haar measure as their probability law.
Besides, we assume without lost of generality that A and B are diagonal:
Ajk = αjδjk, Bjk = βjδjk.
Note that while the probability distributions of eigenvalues of (1.1) and (1.2)
coincide, this is not the case, in general, for entries.
The second class consists of the matrices
W = SU∗TU (1.3)
and
Ŵ = V ∗SV U∗TU. (1.4)
Here we assume that S and T are unitary non-random and diagonal, V and U
are the same as in (1.2).
Let
G(z) = (A + U∗BU − zI)−1 (1.5)
be the resolvent of (1.1). We use the same notation for the resolvents of (1.1)
and (1.2) and for the resolvents of the ensembles (1.3) and (1.4). The complex
variable z will vary over the set Im z 6= 0 for (1.1) and (1.2) and over |z| 6= 1 for
(1.3) and (1.4).
Introduce the normalized eigenvalue counting measure (NCM) of any n × n
Hermitian matrix M as follows:
Nn(Ω) = #{λ(n,M)
l ∈ Ω, l = 1, . . . , n}n−1,
where Ω ⊂ R and
{
λ
(n,M)
l
}
are the eigenvalues of M . Analogously, for any n×n
unitary matrix Q, the NCM is defined for any interval Ω ⊂ T = {z ∈ C : |z| = 1}.
A standard tool to study the fluctuations of eigenvalues are the linear eigen-
value statistics [2–4]
Nn[ϕ] := Trϕ(Hn) =
n∑
l=1
ϕ(λHn
l ) = n
+∞∫
−∞
ϕ(λ)Nn(dλ), (1.6)
where ϕ : R→ C is a measurable and bounded function, e.g.,
Nn [ϕz] = TrG(z), ϕz(λ) = (λ− z)−1. (1.7)
In [3, 4], this study is based on the following two useful propositions:
452 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
Proposition 1.1. If ϕ : R → R, ϕ ∈ L-vector space with the norm ‖ · ‖ and
we have:
• the variance Vn[ϕ] = Var{Nn[ϕ]} admits uniform in the n bound Vn[ϕ] ≤
C||ϕ||2 for all ϕ ∈ L;
• there exists a dense linear manifold L1 ⊂ L such that the Central Limit
Theorem is valid for N ◦
n [ϕ] with ϕ ∈ L1, i.e., if Zn[xϕ] = E
{
eixN ◦
n [ϕ]
}
, then
there exists a continuous quadratic functional V : L1 → R+ such that uniformly
in x on any compact
lim
n→∞Zn[xϕ] = e−x2V [ϕ]/2, ∀ϕ ∈ L1,
then V admits a continuous extension to L and the Central Limit Theorem is
valid for all N ◦
n [ϕ] with ϕ ∈ L.
Proposition 1.2. For any s > 0 and
‖ϕ‖2
s =
+∞∫
−∞
(1 + 2|k|)2s|ϕ̂(k)|2dk, ϕ̂(k) =
1
2π
+∞∫
−∞
eikxϕ(x)dx,
we have Var{Nn[ϕ]} ≤
≤ C ‖ϕ‖s
+∞∫
0
dye−yy2s−1
+∞∫
−∞
Var{TrG(x + iy)}dx.
These tools allow one to prove the Central Limit Theorem for sufficiently
wide classes of test functions of the linear eigenvalue statistics, provided that it
is proved for some special classes, e.g., analytic or even ϕz of (1.7), for a certain
domain of z’s. Accordingly, we will first prove the Central Limit Theorem for the
elements Gjj of the resolvent and then generalize it by using the analogs of the
above propositions. Note that Gjj and Nn[ϕ]jj := ϕ(H)jj (or ϕ(W )jj) are not
the linear eigenvalue statistics (1.6) but just certain statistics of eigenvalues and
eigenvectors. We will thus essentially use the linearity of Nn[ϕ]jj in ϕ.
2. Covariance of Resolvents of (1.1) and (1.2)
We will need the Stieltjes transform
g(z) = n−1TrG =
+∞∫
−∞
Nn(dλ)
λ− z
of the NCM of (1.1).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 453
V. Vasilchuk
Let Nn,A and Nn,B be the Normalized Counting Measures of A and B. We
assume that the measures converge weakly to the probability measures NA and
NB:
lim
n→∞Nn,A = NA, lim
n→∞Nn,B = NB. (2.1)
We will also need the following result obtained in [1, 2]:
Theorem 2.1. Consider the n×n random Hermitian matrices (1.1) (or (1.2)),
where A and B are non-random, satisfying condition (2.1). Then there exists a
non-random measure N such that the Normalized Counting Measures of eigen-
values of (1.1) (or (1.2)) converge weakly with probability 1 to N .
Moreover, the Stieltjes transform
f(z) =
+∞∫
−∞
N(dλ)
λ− z
, Im z 6= 0,
of N is a unique solution of the system
f(z) = fA(hB(z)),
f(z) = fB(hA(z)),
f−1(z) = z − hA(z)− hB(z),
(2.2)
where
fA,B(z) =
+∞∫
−∞
NA,B(dλ)
λ− z
, (2.3)
f is the Stieltjes transform of a probability measure and hA,B are analytic in C\R
functions verifying the conditions
f(z) = −z−1 + o(z−1), hA,B(z) = z + o(z), z →∞. (2.4)
The first result of this paper is as follows:
Theorem 2.2. Consider the random matrices (1.1) and (1.2), where A and
B are non-random, satisfying condition (2.1), and ||A||, ||B|| ≤ M < ∞. Then
we have for Gkk and z1,2 ∈ C\R that
Cov{Gkk(z1), Gkk(z2)} =
1
n
Sn(z1, z2) + rn(z1, z2), (2.5)
where in the case of (1.1),
Sn(z1, z2) =
((GA(hB(z1))−GA(hB(z2)))kk)
2
δf
(
1
δz
− 1
δhB
)
, (2.6)
454 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
and in the case of (1.2),
Sn(z1, z2) =
δf
δz
− f(z1)f(z2), (2.7)
in which
δz = z1 − z2, δf = f(z1)− f(z2),
δhB = hB(z1)− hB(z2), GA(z) = (A− zI)−1,
and the remainder rn(z1, z2) admits the bound
|rn(z1, z2)| ≤ C/n3/2,
where C is independent of n and is finite if min{|Im z1|, |Im z2|} > 0.
R e m a r k 2.3. In the case A = αI, (1.1) and (1.2) coincide, and hence (2.6)
and (2.7) coincide as well. Indeed, we have
GA(hB(z)) =
1
α− hB(z)
I = fA(hB(z))I = f(z)I
and, by the resolvent identity, we have
GA(hB(z1))−GA(hB(z2))
δhB
= GA(hB(z1))GA(hB(z2)) = f(z1)f(z2)I.
Moreover, if also B = βI, then the r.h.s. in (2.6) is zero since
hB(z) = z − β, δhB = δz,
i.e., there is no randomness at all. And this is only the case where the leading
terms (2.6) and (2.7) of the covariances are both zero.
R e m a r k 2.4. The results similar to Theorem 2.2 for various random ma-
trix ensembles with independent matrix entries were recently obtained in [7–9].
In these papers, as well as in Theorem 2.2, the asymptotic regime for the vari-
ance is a classical one, i.e., n−1, but the validness of the Central Limit Theorem
(CLT) strongly depends on the probability distributions of the matrix entries. In
Gaussian case, CLT is valid, otherwise it is not valid in general.
E x a m p l e 2.5. Consider (1.2) with A = B and NA(dλ) = αδ(λ + 1)+
(1− α)δ(λ− 1), 0 ≤ α ≤ 1. Then (2.2) reduces to the quadratic equation
(z2 − 4)f2(z)− 4(2α− 1)f(z)− 1 = 0
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 455
V. Vasilchuk
with the solution
f(z) =
2(2α− 1)
z2 − 4
−
√
4(2α− 1)2 + z2 − 4
z2 − 4
and the resulting measure
N(dλ)=
[
α− 1
2
]
+
δ(λ + 2)+
[
1
2
− α
]
+
δ(λ− 2)+
√
(λ− λ)(λ+ − λ)
π(λ + 2)(2− λ)
χ[λ−,λ+](λ)dλ,
where χ[λ−,λ+] is the indicator of the segment [λ−, λ+], and
λ± = ±2
√
1− (2α− 1)2, [x]+ =
{
x, x > 0;
0, x ≤ 0.
Thus, we have in this case:
S(z1, z2) =
z1 + z2
u(z1)u(z2)
−2(2α− 1) +
1 +
4(2α− 1)2(u(z1) + u(z2))
u(z1)u(z2)
w(z1)
u(z1)
+
w(z2)
u(z2)
−(2(2α− 1)− w(z1))(2(2α− 1)− w(z2))
u(z1)u(z2)
,
where u(z) = z2 − 4, w(z) =
√
4(2α− 1)2 + z2 − 4.
3. Proof of Theorem 2.2
In what follows, by 〈. . .〉 we denote the integral in U (conditional expectation)
with respect to the normalized Haar measure of U(n) and by E {. . .} , the double
integral over U and V (expectation) in (1.2). We denote by a◦ = a − E {a} the
centered random variable. We will use the following two facts [1–3].
Proposition 3.1. Let Hn be the space of the n × n Hermitian matrices and
Φ : Hn → C be a continuously differentiable function. Then we have for any
X ∈ Hn: 〈
Φ′(U∗MU) · [X,U∗MU ]
〉
= 0,
where
[M1,M2] = M1M2 −M1M2.
Proposition 3.2. Let Φ : U(n) → C be a continuously differentiable function.
Then
Var{Φ(Un)} := E{|Φ(Un)|2} − |E{Φ(Un)}|2 ≤ 1
n
n∑
j,k=1
E{|Φ′(Un) · E(j,k)Un|2},
456 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
where {E(j,k)}n
j,k=1 is a canonical basis in the space Mn of all n × n matrices:
E(j,k) = {E(j,k)
pq }n
p,q=1, E
(j,k)
pq = δjpδkq.
P r o o f of Theorem 2.2. We start from the case (1.1). In Proposition 3.1,
choosing Φ =
(
TrE(k,k)G(z1)
)◦
Gac(z2) and using the resolvent identity for the
pair (H, A), we obtain
〈(
TrE(k,k)G(z1)
)◦
(G(z2) [X,U∗BU ]G(z2))ac
〉
+
〈(
TrE(k,k)G(z1) [X,U∗BU ]G(z1)
)
Gac(z2)
〉
= 0.
Then we take X = E(a,b) and apply the operation n−1
n∑
a=1
to the result. This
yields the matrix equality
〈G◦
kk(z1)δn,B(z2)G(z2)〉 = 〈G◦
kk(z1)gn(z2)U∗BUG(z2)〉
+n−1
〈[
U∗BU,G(z1)E(k,k)G(z1)
]
G(z2)
〉
,
where
δn,B = n−1TrU∗BUG. (3.1)
Rewrite U∗BUG by using the resolvent identity for the pair (H,A) as U∗BUG =
I − (A− z)G. After regrouping the terms, we get
fn(z2) (A− hn,B(z2)I) 〈G◦
kk(z1)G(z2)〉 =
1
n
〈[
U∗BU,G(z1)E(k,k)G(z1)
]
G(z2)
〉
+ 〈G◦
kk(z1)g◦n(z2)U∗BUG(z2)〉 −
〈
G◦
kk(z1)δ◦n,B(z2)G(z2)
〉
, (3.2)
where
fn(z) = E{gn(z)}, hn,B(z) = z − E{δn,B(z)}
fn(z)
. (3.3)
In view of ||A||, ||B|| ≤ M, we have the bounds
||G(z)|| ≤ 1
|Im z| , |gn(z)| ≤ 1
|Im z| , fn(z) = −1
z
+ o
(
1
z
)
, (3.4)
|δn,B(z)| ≤ m
(1)
B
|Im z| , m
(k)
A,B = sup
n
+∞∫
−∞
|λ|kNn,A,B(dλ) ≤ Mk,
|gn(z)| ≥ |z|−1
(
1− 2M
|Im z|
)
, |hn,B(z)| ≥ |z|
(
1− M
|Im z| − 2M
)
valid uniformly in n in the domain
Γα,β = {z ∈ C : |Re z| ≤ α|Im z|, |Im z| ≥ β} , β > (2α + 6)max{M,M2} (3.5)
and k = 0, . . . , 4.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 457
V. Vasilchuk
Thus,
|Imhn,B(z)| ≥ β
β − (α + 3)M
β − 2M
, (3.6)
and the matrix A− hB(z)I in (3.2) has the inverse
G̃A(z) = (A− hn,B(z)I)−1 = GA(hn,B(z)) (3.7)
bounded uniformly in n and z ∈ Γα,β:
∥∥∥G̃A(z)
∥∥∥ ≤
(
β
β − (α + 3)M
β − 2M
)−1
. (3.8)
Furthermore, multiplying (3.2) by G̃A(z2) from the left then applying the ope-
ration TrE(k,k)· and regrouping the terms, we obtain the relation
Cov{Gkk(z1), Gkk(z2)} = 〈G◦
kk(z1)Gkk(z2)〉 =
1
n
γ(z1, z2) + R1, (3.9)
where
γ(z1, z2) =
1
fn(z2)
(〈
G̃A(z2)U∗BUG(z1)
〉
kk
〈G(z1)G(z2)〉kk
−
〈
G̃A(z2)G(z1)
〉
kk
〈G(z1)U∗BUG(z2)〉kk
)
, (3.10)
R1 =
1
fn(z2)
(
E{G◦
kk(z1)g◦n(z2)G̃A(z2)U∗BUG(z2)}
− E{G◦
kk(z1)δ◦n,B(z2)G̃A(z2)G(z2)}
+ n−1Cov
{(
G̃A(z2)U∗BUG(z1)
)
kk
, (G(z1)G(z2))kk
}
−n−1Cov
{(
G̃A(z2)G(z1)
)
kk
, (G(z1)U∗BUG(z2))kk
})
. (3.11)
Now the Schwarz inequality for the expectation E{. . .} and (3.8) yields for z,
z1,2 ∈ Γα,β,
|R1| ≤
|z|Var1/2{Gkk(z)}
(
MVar1/2{gn(z)}+ Var1/2{δn,B(z)}
)(
β2 β − (α + 3)M
β − 2M
)−1
+
1
n
|z|
1− 2M
β
(
Var1/2
{(
G̃A(z2)U∗BUG(z1)
)
kk
}
Var1/2 {(G(z1)G(z2))kk}
+Var1/2
{(
G̃A(z2)G(z1)
)
kk
}
Var1/2 {(G(z1)U∗BUG(z2))kk}
)
. (3.12)
458 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
Besides, by Proposition 3.2, we have
Var{Gkk(z)} ≤ 1
n
n∑
j,t=1
E
{∣∣∣TrE(k,k)GU∗
(
E(t,j)B −BE(j,t)
)
UG
∣∣∣
2
}
=
1
n
n∑
j,t=1
E
{∣∣∣(BUGE(k,k)GU∗)jt − (UGE(k,k)GU∗B)tj
∣∣∣
2
}
≤ 4M2
nβ4
(3.13)
and, analogously,
Var
{(
G̃A(z2)U∗BUG(z1)
)
kk
}
≤ 8M4
nβ6
(
β − (α + 3)M
β − 2M
)−2
, (3.14)
Var {(G(z1)G(z2))kk} ≤ 8M2
nβ6
,
Var
{(
G̃A(z2)G(z1)
)
kk
}
≤ 4M2
nβ6
(
β − (α + 3)M
β − 2M
)−2
,
Var {(G(z1)U∗BUG(z2))kk} ≤ 16M4
nβ6
.
It was shown in [1, 2] that
Var{gn(z)},Var{δn,B(z)} = O(n−2). (3.15)
Thus, we have that
R1 = O(n−3/2) (3.16)
uniformly in z ∈ Γα,β.
It was shown in [1, 2] that
〈G(z))〉 = G̃A(z) + O(n−1)I.
Thus, using the resolvent identity, we can write the matrix relations:〈
G̃A(z2)U∗BUG(z1)
〉
= G̃A(z2) 〈U∗BUG(z1)〉
=
z1 − hn,B(z1)
δhn,B
(
G̃A(z1)− G̃A(z2)
)
+ O(n−1)I, (3.17)
〈G(z1)G(z2)〉 =
1
δz
(
G̃A(z1)− G̃A(z2)
)
+ O(n−1)I,
〈
G̃A(z2)G(z1)
〉
= G̃A(z2) 〈G(z1)〉 =
(
G̃A(z1)− G̃A(z2)
)
δhn,B
+ O(n−1)I,
〈G(z1)U∗BUG(z2)〉 =
(
z1G̃A(z1)− z2G̃A(z2)
)
δz
− 〈G(z1)AG(z2)〉+O(n−1)I.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 459
V. Vasilchuk
It was shown in [2] that
〈G(z1)AG(z2)〉 = AG̃A(z1)G̃A(z2)−
δ
(
∆B
f
)
δ∆A
δzδf
G̃A(z1)G̃A(z2) + O(n−1)I,
where
∆B = E {δn,B}, ∆A = E
{
n−1TrAG
}
, δ∆A,B = ∆A,B(z1)−∆A,B(z2),
δ
(
∆B
f
)
=
∆B(z1)
f(z1)
− ∆B(z2)
f(z2)
.
Using this relation, we obtain
〈G(z1)U∗BUG(z2)〉 =
δ∆B
δzδf
(
G̃A(z1)− G̃A(z2)
)
+ O(n−1)I.
Substituting these expressions into (3.9), we obtain (2.5) with (2.6).
Now we pass to the case (1.2). Repeating the same steps, we arrive (cf
(3.2)) to
fn(z2) (V ∗AV − hn,B(z2)I) 〈G◦
kk(z1)G(z2)〉
= fn(z2) 〈G◦
kk(z1)〉
+ n−1
〈[
U∗BU,G(z1)E(k,k)G(z1)
]
G(z2)
〉
+ 〈G◦
kk(z1)g◦n(z2)U∗BUG(z2)〉
− 〈
G◦
kk(z1)δ◦n,B(z2)G(z2)
〉
.
Then, multiplying this relation by G̃V ∗AV (z2) = (V ∗AV − hn,B(z2))−1 from the
left then applying the operation TrE(k,k)· , taking the average E{. . .} of the result
and regrouping the terms, we obtain the relation (cf. (3.9))
Cov{Gkk(z1), Gkk(z2)} = Cov
{
Gkk(z1), (G̃V ∗AV (z2))kk
}
(3.18)
+
1
n
1
fn(z2)
(
E
{
G̃V ∗AV (z2)U∗BUG(z1)
}
kk
E {G(z1)G(z2)}kk
−E
{
G̃V ∗AV (z2)G(z1)
}
kk
E {G(z1)U∗BUG(z2)}kk
)
+ R1,
where R1 can be estimated by (3.12), and by the same arguments (Proposi-
tion 3.2), we have again (3.16) R1 = O(n−3/2). On the other hand, repeating our
procedure for
Φ =
(
TrE(k,k)G̃V ∗AV (z1)
)◦
Gac(z2),
460 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
we obtain
Cov
{
G◦
kk(z1), (G̃V ∗AV (z2))kk
}
= Cov
{
(G̃V ∗AV (z1))kk, (G̃V ∗AV (z2))kk
}
+ R2
= Cov
{
(V ∗G̃A(z1)V )kk, (V ∗G̃A(z2)V )kk
}
+ R2, (3.19)
where
R2 =
1
fn(z2)
(
E{(G̃V ∗AV (z1))◦kkg
◦
n(z2)G̃A(z2)U∗BUG(z2)}
−E{(G̃V ∗AV (z1))◦kkδ
◦
n,B(z2)G̃A(z2)G(z2)}.
Analogously, we obtain
R2 = O(n−3/2).
Besides, it was shown in [3] that
∫
V ∈U(n)
V l1kVl2kdV = n−1δl1l2 ,
∫
V ∈U(n)
V l1kVl2kV l3kVl4kdV = (n(n + 1))−1(δl1l2δl3l4 + δl1l4δl2l3).
Thus, we have
E{(G̃V ∗AV (z))kk} = E{(V ∗G̃A(z)V )kk}
=
1
n
Tr G̃A(z) = f(z)
and
Cov
{
(V ∗G̃A(z1)V )kk, (V ∗G̃A(z2)V )kk
}
=
1
n + 1
(
1
n
Tr G̃A(z1)G̃A(z2)− 1
n
Tr G̃A(z1)
1
n
Tr G̃A(z2)
)
(3.20)
=
1
n
(
δf
δhn,B
− f(z1)f(z2)
)
+ O(n−2).
Moreover, taking the expectation E{...} of (3.17), we obtain
E
{
G̃V ∗AV (z2)U∗BUG(z1)
}
=
(
z1 − hn,B(z1)
δhn,B
δf + O(n−1)
)
I, (3.21)
E {G(z1)G(z2)} =
(
δf
δz
+ O(n−1)
)
I,
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 461
V. Vasilchuk
E
{
G̃V ∗AV (z2)G(z1)
}
=
(
δf
δhn,B
+ O(n−1)
)
I,
E {G(z1)U∗BUG(z2)} =
(
δ∆B
δz
+ O(n−1)
)
I.
Substituting (3.20) and (3.21) into (3.18), we obtain (2.5) with (2.7).
R e m a r k 3.3. It was proved in [5] that the functions hA,B in the system
(2.2) are not only analytic in C\R, but are also the Nevanlinna functions obeying
asymptotics (2.4):
hA,B(z) = z + o(z), z →∞.
Using this, we can carry out the proofs of the above theorem not only for the
region Γα,β, but for the whole C\R. In this case,
|Im hA,B(z)| ≥ |Im z| , z ∈ C\R.
Thus, once we have proved the convergence of (fn, hn,A, hn,B) to the solution of
the limiting system (2.2), we have hn,B(z) → hB(z), z ∈ C\R and, hence, the
matrix G̃A of (3.7) admits the bound
||G̃A(z)|| ≤ C (3.22)
uniformly in z in a compact set K ⊂ C\R and sufficiently large n with an n-
independent C, finite for Im z 6= 0. Since Proposition 3.2 provides the bounds
(3.13) and (3.14) for the variances for any z ∈ C\R, we can estimate all above
remainders for any z ∈ C\R.
4. Central Limit Theorem for the Entries of Functions of (1.1)
and (1.2)
In this section we use (2.6) and (2.7) for the covariance to prove the Central
Limit Theorem for the entries of the resolvent and then for the entries of the
sufficiently smooth function of the matrices (1.1) and (1.2). The first result is
Theorem 4.1. Consider the ensembles (1.1) and (1.2) satisfying the condi-
tions (2.1) and ||A||, ||B|| ≤ M < ∞, then
√
n (Gkk(z)−E {Gkk(z)}) converges
in distribution to the complex Gaussian random variable ξ(z) + iη(z) with zero
mean and the covariances:
Var{ξ(z)} = 2−1Re (S(z, z) + S(z, z)),
Var{η(z)} = −2−1Re (S(z, z)− S(z, z)),
Cov{ξ(z), η(z)} = 2−1Im (S(z, z)− S(z, z)),
where S(z1, z2) is defined in (2.6) and (2.7) for (1.1) and (1.2), respectively.
462 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
P r o o f. The proof follows the proof of the corresponding theorem for the
fluctuations of ng◦n(z) in [2].
Since the proofs for the ensembles (1.1) and (1.2) coincide, let us prove the
theorem only for the ensemble (1.1). Consider Re
√
nG◦
kk(z) and its characteristic
function
Fn(t) = E{ψ(t)}, ψ(t) = exp
{
itRe
√
nG◦
kk(z)
}
. (4.1)
To prove the convergence of Re
√
nG◦
kk(z) to the Gaussian random variable α(z)
with zero mean and variance µ(z) = Var{α(z)}, it suffices to obtain the asymp-
totic relation
d
dt
Fn(t) = −µ(z)tFn(t) + o(1), n →∞, (4.2)
uniformly in t on a compact interval.
Differentiating (4.1) with respect to t, we obtain
d
dt
Fn(t) =
i
√
n
2
(E{ψ(t)G◦
kk(z1)}+ E{ψ(t)G◦
kk(z2)}) ,
where z1 = z, z2 = z. To find E{√nψ(t)G◦
kk(z1)}, we use Proposition 3.1 with
Φ = ψ◦(t)Gac(z1) and repeat the procedure similar to that used in the proof of
the preceding theorem. This leads to the relation
E{√nψ(t)G◦
kk(z1)} =
itFn(t)
2
(γ(z1, z1) + γ(z1, z2)) + O(n−1/2),
where the function γ is defined in (3.10). This relation and the analogous relation
for E{√nψ(t)G◦
kk(z2)} lead to (4.2) with
µ(z) = 4−1 (S(z1, z1) + S(z2, z2) + 2S(z1, z2)) .
Thus, we have proved the theorem for Re
√
nG◦
kk(z). Other assertions can be
proved in a similar way.
Consider now the arbitrary test function ϕ : R→ R. The following analogs of
Propositions 1.1 and 1.2 not for the linear statistics Nn[ϕ], but for the elements
Nn[ϕ]kk = ϕ(H)kk are valid:
Proposition 4.2. If ϕ : R → R, ϕ ∈ L-vector space with the norm ‖ · ‖ and
we have:
• the variance Vn[ϕ]jj = Var{√nNn[ϕ]kk} admits uniform in n bound Vn[ϕ]kk ≤
C||ϕ||2 for all ϕ ∈ L;
• there exists a dense linear manifold L1 ⊂ L such that the Central Limit
Theorem is valid for
√
nN ◦
n [ϕ]kk with ϕ ∈ L1, i.e., if Zn[xϕ] = E
{
eix
√
nN ◦
n [ϕ]kk
}
,
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 463
V. Vasilchuk
then there exists a continuous quadratic functional Vkk : L1 → R+ such that
uniformly in x on any compact
lim
n→∞Zn[xϕ] = e−x2V [ϕ]kk/2, ∀ϕ ∈ L1,
then Vkk admits a continuous extension to L and the Central Limit Theorem is
valid for all
√
nN ◦
n [ϕ]kk with ϕ ∈ L.
The proof of this proposition coincides with the proof of Proposition 1.1 in
[3] since the statistic Nn[ϕ]kk remains linear in ϕ although it depends not only
on the eigenvalues of H but on its eigenvectors too.
Proposition 4.3. For any s > 0 and ‖ϕ‖2
s =
∫ +∞
−∞ (1 + 2|k|)2s|ϕ̂(k)|2dk, we
have Var{Nn[ϕ]jj} ≤
≤ C ‖ϕ‖s
+∞∫
0
dye−yy2s−1
+∞∫
−∞
Var{Gjj(x + iy)}dx.
The proof of this proposition follows the proof of Proposition 1.2 from [4] with
one simple modification: we consider Var{Nn[ϕ]jj} as a bounded quadratic form
for any fixed n generating a positive self-adjoint operator in a proper space. We
will present below in all details the proof of the analog of this proposition for the
unitary case.
Following [4], we consider the test function ϕ : R→ R from the space Hs(R)
possessing the norm
‖ϕ‖2
s =
+∞∫
−∞
(1 + 2|p|)2s|ϕ̂(p)|2dp, s > 2.
Theorem 4.4. Consider the ensembles (1.1) and (1.2) satisfying the condition
(2.1) and ||A|, ||B|| ≤ M < ∞, and hence supp N being compact, χN (λ) being
its indicator; the function ϕ : R→ R, ϕ ∈ H2+ε(R), ε > 0 and Nn[ϕ]jj being the
corresponding statistics. Then the random variable
√
nN ◦
n [ϕ]kk =
√
nNn[ϕ]kk −
√
nE{Nn[ϕ]kk}
converges in distribution to the Gaussian random variable with zero mean and
the variance
V [ϕ] = lim
η1,η2↓0
1
π2
+∞∫
−∞
+∞∫
−∞
ϕ(λ1)ϕ(λ2) (I1,2 · S) (λ1+iη1, λ2+iη2)χN (λ1)χN (λ2)dλ1dλ2,
464 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
where
(I1,2 · S) (z1, z2) := −Re (S(z1, z2)− S(z1, z2))
and S(z1, z2) is defined in (2.6) and (2.7) for (1.1) and (1.2), respectively.
P r o o f. First of all, let us note that the bound (3.13) implies
Var{Gkk(x + iy)} ≤ 1
n
C
(dist{x + iy, supp NHn})4
≤ 1
n
C
(y2 + (dist{x, supp NHn})2)2
.
Thus, due to the fact that supp NHn ⊂ [−2M, 2M ] and Proposition 4.3, we have
Var{√nNn[ϕ]kk} ≤ C ‖ϕ‖
2+ε
+∞∫
0
dye−yy3+2ε
+∞∫
−∞
nVar{Gkk(x + iy)}dx
≤ C ‖ϕ‖
2+ε
+∞∫
0
dye−yy−1+2εdx ≤ C ‖ϕ‖
2+ε
.
Then, according to Proposition 4.2, it suffices to prove the theorem for any
dense set in H2+ε([−2M, 2M ]), say, for any ϕ analytic in some domain in C
including [−2M, 2M ], e.g., for a polynomial ϕ. Note that due to the Cauchy
theorem,
√
nN ◦
n [ϕ]kk =
√
n
n∑
l=1
(ϕkk(Hn)−E {ϕkk(Hn)})
=
√
n
2πi
∫
Γ
ϕ(z) (−Gkk(z) + E {Gkk(z)}) dz
= −
√
n
2πi
∫
Γ
ϕ(z)G◦
kk(z)dz,
where Γ is any closed contour in the complex plane encircling the segment
[−2M, 2M ] in the real axis. Define the characteristic function
Zn(x) = E {en(x)} , x ∈ R,
where
en(x) = eix
√
nN ◦
n [ϕ]kk = exp
−
√
nx
2π
∫
Γ
ϕ(z)G◦
kk(z)dz
.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 465
V. Vasilchuk
Since Zn(0) = 1 and
en(x) = 1 +
x∫
0
e′n(y)dy, Zn(x) = 1 +
x∫
0
Z ′n(y)dy, (4.3)
it suffices to prove that there exist the subsequences {Znj (x)} and {Z ′nj
(x)} that
converge uniformly on any finite interval, and
lim
nj→∞
Znj (x) = Z(x), lim
nj→∞
Z ′nj
(x) = −xV [ϕ]Z(x).
Besides, due to the Cauchy theorem,
d
dx
en(x) = −
√
n
2π
en(x)
∫
Γ
ϕ(z)G◦
kk(z)dz (4.4)
= −
√
n
2π
∫
Γ1
ϕ(z1)en(x)G◦
kk(z1)dz1,
Z ′n(x) = −
√
n
2π
∫
Γ1
ϕ(z1)E {e◦n(x)Gkk(z1)} dz1,
where we choose the contour Γ1 ⊂ DM in the domain
DM = {z ∈ C : ρ = min dist(z, [−2M, 2M ]) > 4M}.
To find E {e◦n(x)Gkk(z1)} in both cases (1.1) and (1.2), we apply the same pro-
cedure as in the previous section and obtain the analog of (3.9) for the case
(1.1):
Cov{e◦n(x), Gkk(z1)} = 〈e◦n(x)Gkk(z1)〉 (4.5)
= −xZn(x)√
n2π
∫
Γ2
ϕ(z2)γ(z2, z1)dz2 +
√
nR3,
where
R3 =
1
E{gn(z1)}
(
E{e◦n(x)g◦n(z1)G̃A(z1)U∗BUG(z1)}
−E{e◦n(x)δ◦n,B(z1)G̃A(z1)G(z1)}+
x√
n
∫
Γ2
ϕ(z2)R4(z2, z1)dz2
)
and
R4(z2, z1) = E
{
en(x)
(
G̃A(z1)U∗BUG(z2)
)◦
kk
(G(z2)G(z1))
◦
kk
}
−E
{
en(x)
(
G̃A(z1)G(z2)
)◦
kk
(G(z2)U∗BUG(z1))
◦
kk
}
466 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
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+ E
{
en(x)
(
G̃A(z1)U∗BUG(z2)
)◦
kk
}
E {G(z2)G(z1)}kk
−E
{
en(x)
(
G̃A(z1)G(z2)
)◦
kk
}
E {G(z2)U∗BUG(z1)}kk
+ E
{
G̃A(z1)U∗BUG(z2)
}
kk
E {en(x) (G(z2)G(z1))
◦
kk}
−E
{
G̃A(z1)G(z2)
}
kk
E {en(x) (G(z2)U∗BUG(z1))
◦
kk} ,
where the contour Γ2 ⊂ DM . Besides, for z ∈ DM , we have the bounds
|δAn,Bn(z)| ≤ 1
4
, |gn(z)| ≥ 1
2|z| , min dist(hAn,Bn(z), [−M,M ]) ≥ 2M,
||Gn(z)|| ≤ 1
4M
, ||GAn,Bn(hBn,An(z))|| ≤ 1
2M
, |en(x)| ≤ 1.
Moreover, by using (3.12)–(3.15), it can be shown that
R4 = O(n−1/2)
uniformly for z1,2 ∈ Ω, Ω-compact, Ω ⊂ DM . Thus, we obtain
R3 = O(n−1/2)
uniformly in x on any finite interval and in z1,2 ∈ Ω, Ω-compact, Ω ⊂ DM . Then,
using (4.5), we obtain
√
nE{e◦n(x)Gkk(z1)} =
xZn(x)
2π
∫
Γ2
ϕ(z2)γ(z2, z1)dz2 + O(n−1/2)
=
xZn(x)
2π
∫
Γ2
ϕ(z2)S(z1, z2)dz2 + O(n−1/2).
uniformly in x on any finite interval. Substituting this into (4.4), we obtain
Z ′n(x) = −xZn(x)
4π
∫
Γ1
∫
Γ2
ϕ(z1)ϕ(z2)S(z1, z2)dz1dz2 + O(n−1/2),
uniformly in x on any finite interval in view of the finiteness of the contours Γ1,2,
which completes the proof due to the analyticity of ϕ(z1)ϕ(z2)S(z1, z2) in z1,2 for
z1,2 ∈ C\[−2M, 2M ].
The case (1.2) can be treated in a similar way.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 467
V. Vasilchuk
5. Covariance of Resolvents of (1.3) and (1.4)
Now we prove the analogous results for the product. Define Nn,S and Nn,T ,
the NCMs of S and T for any interval Ω ⊂ T = {z ∈ C : |z| = 1},
Nn,S(Ω) = #{s(n)
l ∈ Ω, l = 1, . . . , n}n−1,
Nn,T (Ω) = #{t(n)
l ∈ Ω, l = 1, . . . , n}n−1. (5.1)
We assume that the measures converge weakly to the probability measures NS
and NT :
lim
n→∞Nn,S = NS , lim
n→∞Nn,T = NT . (5.2)
As in the additive case, there is the way to express the NCM of the product in
terms of NCMs of the factors, using, for example, the following result [6]:
Theorem 5.1. If S and T satisfy the condition (5.2), then the NCM Nn of
(1.3) converges weakly with probability 1 to the non-random probability measure
N , and the Stieltjes transform
f(z) =
∫
T
N(dµ)
µ− z
(5.3)
of N is a unique solution of the system
∆T (z) = fS
(
z∆S(z)
1 + zf(z)
)
,
∆S(z) = fT
(
z∆T (z)
1 + zf(z)
)
,
f(z)(1 + zf(z)) = ∆S(z)∆T (z),
(5.4)
where f(z), ∆S(z), ∆T (z) are analytic for the |z| < 1 functions verifying the
conditions
∆S(0) = m
(1)
T , ∆T (0) = m
(1)
S . (5.5)
Despite its bulkiness, when compared with (2.2) in the additive case, the
system (5.4) gives a useful observation:
R e m a r k 5.2. (Divisors of the uniform distribution on T.) If we consider
(5.4) as the definition of the free multiplicative convolution NS £ NT = N of the
measures on T, we can use it to obtain all divisors of uniform distribution on T
with respect to £. Indeed, for the N -uniform distribution, we have f(z) ≡ 0 and
∆T (z) = fS(z∆S(z)),
∆S(z) = fT (z∆T (z)),
∆S(z)∆T (z) = 0.
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Thus, where are only two possibilities: we have the purely trivial solution
(f(z) ≡ 0, ∆S(z) ≡ 0, ∆T (z) ≡ 0), which implies m
(1)
S = m
(1)
T = 0, and at least
one of the measures (say, NS) is uniform. In the last case, 0 ≡ ∆T (z) = fS(z) =
m
(1)
S , ∆S(z) = fT (0) ≡ m
(1)
T (6= 0 in general) and where are no more options.
Thus we have to distinguish the cases f(z) ≡ 0 and f(z) 6= 0.
Theorem 5.3. If S and T are non-random satisfying the condition (5.2) and
f(z) ≡ 0, then we have for Gkk and |z1,2| < 1,
Cov{Gkk(z1), Gkk(z2)} = n−1Sn(z1, z2) + rn(z1, z2),
where
• in the case of (1.3) and 0 ≡ ∆S,T (z),
Sn(z1, z2) =
f
′
T (0)
1− z1z2f
′
S(0)f ′T (0)
(S∗kk)
2, (5.6)
• for 0 ≡ ∆T (z) = fS(z), ∆S(z) = m
(1)
T ,
Sn(z1, z2) =
(
SGS
(
m
(1)
T z1
)
GS
(
m
(1)
T z2
))
kk
×
f
′
T (0)S∗kk −
(
m
(1)
T
)2
δz
(
GS
(
m
(1)
T z1
)
−GS
(
m
(1)
T z2
))
kk
;
(5.7)
• in the case of (1.3) and 0 ≡ ∆S(z) = fT (z) and in the case of (1.4),
Sn(z1, z2) = 0 (5.8)
and
|rn(z1, z2)| ≤ C/n−3/2.
P r o o f. We use the scheme of the proof of Theorem 2.2: first we consider
the ensemble (1.3) and then the ensemble (1.4). In Proposition 3.1, taking Φ =(
TrE(k,k)G(z1)
)◦
Gac(z2) and using the resolvent identity for the pair (H, A), we
obtain
〈(
TrE(k,k)G(z1)
)◦
(G(z2)S [X, U∗TU ]G(z2))ac
〉
+
〈(
TrE(k,k)G(z1)S [X, U∗TU ] G(z1)
)
Gac(z2)
〉
= 0.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 469
V. Vasilchuk
Then, taking X = E(a,b) and applying the operation n−1
n∑
a=1
to the result, we
obtain the matrix equality
〈G◦
kk(z1)(1 + z2gn(z2))G(z2)〉 = 〈G◦
kk(z1)δn,S(z2)U∗TUG(z2)〉
+n−1
〈[
U∗TU,G(z1)E(k,k)G(z1)S
]
G(z2)
〉
,
where
δn,S = n−1TrSG. (5.9)
Rewrite G by using the resolvent identity for the pair (W, I) as G = z−1(SU∗TUG
−I). After regrouping the terms, we have
1 + z2fn(z2)
z2
(S − ρn,S(z2)I) 〈G◦
kk(z1)U∗TUG(z2)〉
= n−1
〈[
U∗TU,G(z1)E(k,k)G(z1)S
]
G(z2)
〉
+
〈
G◦
kk(z1)δ◦n,B(z2)U∗TUG(z2)
〉
− z2 〈G◦
kk(z1)g◦n(z2)G(z2)〉 , (5.10)
where
fn(z) = E{gn(z)}, ρn,S(z) =
zE{δn,S(z)}
1 + zfn(z)
. (5.11)
Besides, we have the bounds
||G(z)|| ≤ 1
|1− |z|| , |gn(z)| ≤ 1
|1− |z|| , |δn,S(z)| ≤ 1
|1− |z|| , |z| 6= 1,
|ρn,B(z)| ≤ |z|
1− 2|z| , |z| <
1
2
, |ρn,B(z)| ≤ 1
2
, |z| < 1
4
. (5.12)
Thus the matrix S − ρn,S(z2)I in (5.10) has the inverse
G̃S(z) = (S − ρn,S(z)I)−1 = GS(ρn,S(z))
uniformly in n bounded ||G̃S(z)|| ≤ 2 for |z| < 1/4. Next, multiplying (5.10) by
G̃S(z2) from the left, we obtain the relation
〈G◦
kk(z1)U∗TUG(z2)〉
=
1
n
z2
1 + z2fn(z2)
〈
G̃S(z2)
[
U∗TU,G(z1)E(k,k)G(z1)S
]
G(z2)
〉
+
〈
G◦
kk(z1)δ◦n,B(z2)G̃S(z2)U∗TUG(z2)
〉
− z2
〈
G◦
kk(z1)g◦n(z2)G̃S(z2)G(z2)
〉
.
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Multiplying this relation by 1
z2
S from the left, regrouping the terms according to
the resolvent identity and applying then the operation TrE(k,k)· , we obtain
Cov{Gkk(z1), Gkk(z2)} = 〈G◦
kk(z1)Gkk(z2)〉 =
1
n
γ(z1, z2) + R5, (5.13)
where
γ(z1, z2) =
1
1 + z2fn(z2)
(〈
SG̃S(z2)U∗TUG(z1)
〉
kk
〈G(z1)SG(z2)〉kk
−
〈
SG̃S(z2)G(z1)
〉
kk
〈G(z1)SU∗TUG(z2)〉kk
)
, (5.14)
R5 =
1
1 + z2fn(z2)
(
E{G◦
kk(z1)δ◦n,B(z2)SG̃S(z2)U∗BUG(z2)}
− z2E{G◦
kk(z1)g◦n(z2)G̃A(z2)G(z2)}
+ n−1Cov
{(
SG̃S(z2)U∗TUG(z1)
)
kk
, (G(z1)SG(z2))kk
}
−n−1Cov
{(
SG̃S(z2)G(z1)
)
kk
, (G(z1)SU∗TUG(z2))kk
})
. (5.15)
Now the Schwarz inequality for the expectation E{. . .}, (5.12) and the bound
||G̃S(z)|| ≤ 2 for |z| ≤ 1/4 yield for |z|, |z1,2| ≤ 1/4,
|R5| ≤ 16
3
Var1/2{Gkk(z)}
(
Var1/2{gn(z)}+ Var1/2{δn,B(z)}
)
+
1
n
3
2
(
Var1/2
{(
SG̃S(z2)U∗TUG(z1)
)
kk
}
Var1/2 {(G(z1)SG(z2))kk}
+Var1/2
{(
SG̃S(z2)G(z1)
)
kk
}
Var1/2 {(G(z1)SU∗TUG(z2))kk}
)
. (5.16)
Besides, by using Proposition 3.2, we have
Var{Gkk(z)} ≤ 1
n
n∑
j,t=1
E
{∣∣∣TrE(k,k)GSU∗
(
E(t,j)T − TE(j,t)
)
UG
∣∣∣
2
}
=
1
n
n∑
j,t=1
E
{∣∣∣(TUGE(k,k)GSU∗)jt − (UGE(k,k)GSU∗T )tj
∣∣∣
2
}
≤ 4
n|1− |z||4 , |z| 6= 1, (5.17)
and, analogously,
Var {(G(z1)SG(z2))kk} ≤ 8
n|1− |z||4 , |z| 6= 1,
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 471
V. Vasilchuk
Var {(G(z1)SU∗TUG(z2))kk} ≤ 8
n|1− |z||4 , |z| 6= 1,
Var
{(
SG̃S(z2)U∗TUG(z1)
)
kk
}
≤ 8
n|1− |z||4 , |z| ≤ 1
4
,
Var
{(
SG̃S(z2)G(z1)
)
kk
}
≤ 8
n|1− |z||4 , |z| ≤ 1
4
.
On the other hand, by using Proposition 3.2, in the same way, for |z| 6= 1 we
obtain
Var{gn(z)},Var{δn,B(z)} = O(n−2).
Thus, we have that
R5 = O(n−3/2)
uniformly in |z|, |z1,2| ≤ 1/4. Besides, in [6] it was proved that
〈U∗TUG(z)〉 = G̃S(z) + O(n−1) (5.18)
uniformly in |z| ≤ 1/4. Due to this relation and Proposition 7.1, for 0 ≡ f(z) =
∆S,T (z) we have
〈
SG̃S(z2)U∗TUG(z1)
〉
kk
=
(
SG̃S(z2)G̃S(z1)
)
kk
+ O(n−1)
=
(
SG̃2
S(0)
)
kk
+ O(n−1) = (S∗)kk ,
〈G(z1)SG(z2)〉kk =
f
′
T (0)
1− z1z2f
′
S(0)f ′T (0)
S∗kk + O(n−1),
〈
SG̃S(z2)G(z1)
〉
kk
=
ρn,S(z1)
z1
(
SG̃S(z2)G̃S(z1)
)
kk
+ O(n−1)
= O(n−1).
Substituting this into (5.14), we obtain (5.6). Analogously, in the case 0 ≡ f(z) =
∆T (z) = fS(z), ∆S(z) = m
(1)
T , we have
〈
SG̃S(z2)U∗TUG(z1)
〉
kk
=
(
SGS
(
m
(1)
T z2
)
GS
(
m
(1)
T z1
))
kk
+ O(n−1),
〈G(z1)SG(z2)〉kk = f
′
T (0)S∗kk + O(n−1),
〈
SG̃S(z2)G(z1)
〉
kk
= m
(1)
T
(
SGS
(
m
(1)
T z2
)
GS
(
m
(1)
T z1
))
kk
+ O(n−1),
〈G(z1)SU∗TUG(z2)〉kk =
m
(1)
T
δz
(
z1GS
(
m
(1)
T z1
)
−z2GS
(
m
(1)
T z2
))
kk
+ O(n−1)
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and then we obtain (5.7). In the case 0 ≡ f(z) = ∆S(z) = fT (z), we have
〈G(z1)SG(z2)〉kk = O(n−1),〈
SG̃S(z2)G(z1)
〉
kk
= O(n−1)
and (5.8).
Now we turn to the ensemble (1.4). Following the scheme of the proof of
Theorem 2.2 we arrive (cf. (5.10)) to the relation
1 + z2fn(z2)
z2
(V ∗SV − ρn,S(z2)I) 〈G◦
kk(z1)U∗TUG(z2)〉
=
1 + z2fn(z2)
z2
〈G◦
kk(z1)〉
+ n−1
〈[
U∗TU,G(z1)E(k,k)G(z1)S
]
G(z2)
〉
+
〈
G◦
kk(z1)δ◦n,B(z2)U∗TUG(z2)
〉− z2 〈G◦
kk(z1)g◦n(z2)G(z2)〉 .
Multiplying this relation first by G̃V ∗SV (z2) = (V ∗SV − ρn,S(z2))−1 and then by
z−1
2 V ∗SV from the left, applying then the operation TrE(k,k)·, taking the average
E{. . .} of the result and regrouping the terms, we obtain the relation
Cov{Gkk(z1), Gkk(z2)} =
ρn,S(z2)
z2
Cov
{
Gkk(z1), (G̃V ∗SV (z2))kk
}
+
1
n
1
1 + z2fn(z2)
(
E
{
V ∗SV G̃V ∗SV (z2)U∗TUG(z1)
}
kk
E {G(z1)V ∗SV G(z2)}kk
−E
{
V ∗SV G̃V ∗SV (z2)G(z1)
}
kk
E {G(z1)V ∗SV U∗TUG(z2)}kk
)
+ R6, (5.19)
where R6 = O(n−3/2). Using the same arguments as in the proof of Theorem
2.2, we obtain
ρn,S(z2)
z2
Cov
{
Gkk(z1), (G̃V ∗SV (z2))kk
}
=
ρn,S(z1)ρn,S(z2)
z1z2
Cov
{
(G̃V ∗SV (z1))kk, (G̃V ∗SV (z2))kk
}
+ O(n−3/2)
=
ρn,S(z1)ρn,S(z2)
nz1z2
(
1
n
Tr G̃S(z1)G̃S(z2)−∆T (z1)∆T (z2)
)
+ O(n−3/2) = O(n−3/2)
(5.20)
for 0 ≡ f(z). Moreover, due to (5.18) and Proposition 7.1, we have
E {G(z1)V ∗SV G(z2)}kk = O(n−1),
E {G(z1)V ∗SV U∗TUG(z2)}kk =
δ(zf)
δz
+ O(n−1) = O(n−1)
for 0 ≡ f(z). Thus we finally arrive to (5.8).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 473
V. Vasilchuk
Theorem 5.4. If S and T are non-random, satisfying condition (5.2) and
f(z) 6= 0, then for Gkk and |z1,2| < 1 we have
Cov{Gkk(z1), Gkk(z2)} = n−1Sn(z1, z2) + rn(z1, z2),
where
• in the case of (1.3),
Sn(z1, z2) =
((ρn,S(z1)GS(ρn,S(z1))− ρn,S(z2)GS(ρn,S(z2)))kk)2
(z1 − z2)(z1f(z1)− z2f(z2))
× z2ρn,S(z1)− z1ρn,S(z2))
z1z2(ρn,S(z1)− ρn,S(z2))
, (5.21)
• in the case of (1.4), we have (cf. (2.7))
Sn(z1, z2) =
fn(z1)− fn(z2)
z1 − z2
− fn(z1)fn(z2) (5.22)
and |rn(z1, z2)| ≤ C/n−3/2.
R e m a r k 5.5. (cf. Remark 2.3) Note that in the case S = σI, when (1.3)
and (1.4) coincide, the expressions (5.21) and (5.22) coincide as well. Therefore
we have
ρn,S(z) =
σzf(z)
1 + zf(z)
, GS(ρn,S(z)) =
1
σ − ρn,S(z)
I =
1 + zf(z)
σ
I.
E x a m p l e 5.6. (cf. Example 2.5) Consider the case S = T and the measure
NS(dλ) = αδ(λ + 1) + (1 − α)δ(λ− 1), 0 ≤ α ≤ 1 for the ensemble (1.4), again
having two point masses α and 1− α at the points −1 and 1, respectively. Then
(2.2) reduces to the quadratic equation
(z2f2(z) + zf(z))(z−1 + z − 2)− (2α− 1)2 = 0
with the solution
zf(z) = −1
2
−
√
(z−1 + z − 2)2 + 4(2α− 1)2(z−1 + z − 2)
2(z−1 + z − 2)
and the resulting measure
N(dθ) = |2α− 1|δ(λ− 1) +
1
π
√
1− 2(2α− 1)2 − cos θ
1− cos θ
χ[θ+,θ−](θ)dθ,
where χ[θ+,θ−] is the indicator of the segment [θ−, θ+] and
θ+ = arccos(1− 2(2α− 1)2), θ− = 2π − arccos(1− 2(2α− 1)2).
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Thus, for this case we have
S(z1, z2) =
1
2z1z2
+
z1 + z2 + 4(2α− 1)2
z2
1 + z1z2 + z2
2 − 2(z1 + z2) + 1
u(z1)u(z2)
2z1z2
(
w(z1)
u(z1)
+
w(z2)
u(z2)
)
− 1
4
(
1
z1
+
w(z1)
u(z1)
)(
1
z2
+
w(z2)
u(z2)
)
,
where u(z) = (z − 1)2, w(z) =
√
(z−1 + z − 2)2 + 4(2α− 1)2(z−1 + z − 2).
P r o o f of Theorem 5.4. Following the proof of the previous theorem for
the ensemble (1.3) up to the relation (5.13), we use (5.18) and Proposition 7.1 to
obtain
〈
SG̃S(z2)U∗TUG(z1)
〉
kk
=
(
SG̃S(z2)G̃S(z1)
)
kk
+ O(n−1)
=
(
ρn,S(z1)G̃S(z1)− ρn,S(z2)G̃S(z2)
)
kk
ρn,S(z1)− ρn,S(z2)
+ O(n−1),
〈G(z1)SG(z2)〉kk =
(
ρn,S(z1)G̃S(z1)− ρn,S(z2)G̃S(z2)
)
kk
δ∆S
δzδ(zf)
+ O(n−1),
〈
SG̃S(z2)G(z1)
〉
kk
=
ρn,S(z1)
z1
(
ρn,S(z1)G̃S(z1)− ρn,S(z2)G̃S(z2)
)
kk
ρn,S(z1)− ρn,S(z2)
+ O(n−1),
〈G(z1)SU∗TUG(z2)〉kk =
(z1 〈G(z1)〉 − z2 〈G(z2)〉)kk
δz
+ O(n−1)
=
(
ρn,S(z1)G̃S(z1)− ρn,S(z2)G̃S(z2)
)
kk
δz
+ O(n−1).
Substituting this into (5.13), we obtain the assertion of the theorem for the
ensemble (1.3). To complete the proof for the ensemble (1.4), we follow the
path of the proof of the previous theorem up to the relations (3.9)–(5.20) and
substitute into (3.9)–(5.20) the relations below:
E
{
V ∗SV G̃V ∗SV (z2)U∗TUG(z1)
}
kk
=
δ(zf)
ρn,S(z1)− ρn,S(z2)
+ O(n−1),
E {G(z1)V ∗SV G(z2)}kk =
δ∆S
δz
+ O(n−1),
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 475
V. Vasilchuk
E
{
V ∗SV G̃V ∗SV (z2)G(z1)
}
kk
=
ρn,S(z1)
z1
δ(zf)
δz
+ O(n−1),
E {G(z1)SV ∗SV U∗TUG(z2)}kk =
δ(zf)
δz
+ O(n−1).
After simple algebra we obtain the assertion of the theorem for the ensemble
(1.4).
6. Central Limit Theorem for the Entries of Functions
of (1.3) and (1.4)
In the same way as in Section 3, we will prove the Central Limit Theorems
for the elements of the statistic function ϕ of the matrices (1.3) and (1.4) using
the explicit formulas (5.21) and (5.22), respectively. To apply the scheme of the
proof of Theorem 4.4 with the dense set of test functions, we have to satisfy
the conditions of Proposition 4.2. Thus we need the analog of Proposition 4.3
below to have a priori bound for the variance of the statistic. Let us consider the
function ϕ : T→ C from the Hilbert space Hs(T), s ≥ 0 with the norm
||ϕ||2s =
+∞∑
k=−∞
(1 + 2|k|)2s|ϕ̂k|2, ϕ̂k =
1
2π
π∫
−π
ϕ(eiθ)e−ikθdθ.
Proposition 6.1. For any ϕ ∈ Hs(T), s > 0, and any random unitary W
we have the bound
Var{Nn[ϕ]jj} ≤ Cs||ϕ||2s
+∞∫
0
dte−tt2s−1
π∫
−π
Var{ReQjj(e−teiθ)}dθ, (6.1)
where Q(z) = (W + zI)(W − zI)−1, |z| < 1.
P r o o f. We just follow the proof of the analogous proposition for the
Hermitian case from [4]. Consider the operator Ds : Hs(T) → L2(T) defined for
any ψ ∈ Hs(T),
(̂Dsψ)k = (1 + 2|k|)sψ̂k.
For fixed n, Var{Nn[ϕ]jj} is a bounded quadratic form in the Hilbert space
Hs(T) with the inner product (u, v)s = (Dsu,Dsv), where (·, ·) denotes the inner
product in L2(T). Then there exists a positive self-adjoint operator Vjj which
defines the quadratic form
Var{Nn[ϕ]jj} = (Vjjϕ,ϕ) = Tr(ΠϕVjjΠϕ),
476 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
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where Πϕ is the modified projection operator Πϕu = ϕ(u, ϕ)/||ϕ||L2 . Besides, we
have
Tr(ΠϕVjjΠϕ) = Tr(ΠϕDsD
−1
s VjjD
−1
s DsΠϕ) ≤ ||DsΠϕ||2Tr(D−1
s VjjD
−1
s ),
where ||DsΠϕ|| ≤ ||Dsϕ||L2 = ||ϕ||s. On the other hand, for any u, v ∈ L2(T), we
have
Γ(2s)(D−2
s u, v) = Γ(2s)
+∞∑
k=−∞
(1 + 2|k|)−2sûkv̂k
=
+∞∫
0
dte−tt2s−1
+∞∑
k=−∞
(
e−t
)2|k|
ûkv̂k
=
+∞∫
0
dte−tt2s−1 (Pe−t ∗ u, Pe−t ∗ v)
=
+∞∫
0
dte−tt2s−1
π∫
−π
dθ
π∫
−π
π∫
−π
Pe−t(θ − λ)Pe−t(θ − µ)u(λ)v(µ)dλdµ,
where Γ denotes the Γ-function, ·*· means the convolution of functions, and Pr(θ)
is the Poisson kernel with the parameter 0 ≤ r < 1,
Pr(θ) =
1− r2
1− 2r cos θ + r2
= Re
1 + reiθ
1− reiθ
.
This implies the explicit form of the integral kernel of Γ(2s)(D−2
s u, v),
Γ(2s)D−2
s (λ, µ) =
+∞∫
0
dte−tt2s−1
π∫
−π
dθPe−t(θ − λ)Pe−t(θ − µ)
and
Γ(2s)Tr(D−1
s VjjD
−1
s ) =
+∞∫
0
dte−tt2s−1
π∫
−π
dθ (VjjPe−t(θ − .), Pe−t(θ − .))
=
+∞∫
0
dte−tt2s−1
π∫
−π
dθVar{Nn[Pe−t(θ − .)]jj}
=
+∞∫
0
dte−tt2s−1
π∫
−π
Var{ReQjj(e−teiθ)}dθ,
which completes the proof.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 477
V. Vasilchuk
Before formulating the main result of this section, we should note that al-
though the solution of the system (5.4) and the formulas (5.21), (5.22) are defined
only for |z| < 1, in the case f(z) 6= 0, we can always define them also for |z| > 1.
Indeed, using the new variables
ψ(z) =
zf(z)
1 + zf(z)
,
ρS,T (z) =
z∆S,T (z)
1 + zf(z)
,
one can rewrite the system (5.4) in the form
ψ(z) = ψS(ρS(z)),
ψ(z) = ψT (ρT (z)),
zψ(z) = ρS(z)ρT (z),
where
ψS,T (z) =
zfS,T (z)
1 + zfS,T (z)
.
On other hand, under the condition f(z) 6= 0, |z| < 1, using the integral repre-
sentation (5.3), one can define ψ(z) for |z| > 1 via the formula
ψ
(
1
z̄
)
=
1
ψ(z)
.
Then, using the system above, we obtain
ρS,T
(
1
z̄
)
=
1
ρS,T (z)
.
Thus, in the case f(z) 6= 0, the system above and hence the system (5.4) are valid
for any |z| 6= 1. Moreover, since the corresponding bounds in (5.12) depend in
fact on the distance form z to the unit circle, then Theorem 5.4 with the formulas
(5.21), (5.22) is also valid for any |z1,2| 6= 1.
The main result of this section is as follows:
Theorem 6.2. If S and T are non-random, satisfying condition (5.2) and
f(z) 6= 0, and the test function ϕ : T→ R, ϕ ∈ H2+ε(T), ε > 0, then
√
nN ◦
n [ϕ]kk =
√
nNn[ϕ]kk −
√
nE{Nn[ϕ]kk}
converges in distribution to the Gaussian random variable with zero mean and
the variance
V [ϕ] = lim
r1,r2↑1
1
4π2
π∫
−π
π∫
−π
ϕ(eiθ1)ϕ(eiθ2) (R1,2 · T ) (r1e
iθ1 , r2e
iθ1)dθ1dθ2,
478 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
where
T (z1, z2) : = z1z2S(z1, z2),
(R1,2 · T ) (z1, z2) : = T (z1, z2)− T
(
z1,
1
z2
)
− T
(
1
z1
, z2
)
+ T
(
1
z1
,
1
z2
)
,
and S(z1, z2) is defined in (5.21) and (5.22) for (1.1) and (1.2), respectively.
P r o o f. As in the proof of Theorem 4.4, we prove the theorem for some
dense set in H2+ε(T) (say, trigonometric polynomials) and then extend it on the
whole H2+ε(T) by using Propositions 6.1 and 4.2. The procedure is the same
with the difference in the representation of the analytic in C\{0} test function
ϕ via the contour integral by the Cauchy formula. We start with the contour
Γ = Γr ∪ Γ+
β ∪ Γ1/r ∪ Γ−r (see Fig. 1.(a)) encircling the spectrum of the ensemble
(1.3) (or (1.4)) for any realization.
Fig. 1. Contours (a) and (b)
Of course, this contour has a realization-dependent part (we choose the angle
β such that the contour Γ lies outside the eigenvalues for each realization), but
we can cancel the integrals over Γ+
β and Γ−r since they are the same contour
integrals in opposite directions. Thus we obtain the integral over the realization-
independent contour Γr ∪ Γ1/r (see Fig. 1.(b))
√
nN ◦
n [ϕ]kk = −
√
n
2πi
∫
Γr∪Γ1/r
ϕ(z)G◦
kk(z)dz
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 479
V. Vasilchuk
=
√
n
2π
π∫
−π
(
ϕ
(
reiθ
)
reiθG◦
kk
(
reiθ
)
− ϕ
(
eiθ
r
)
eiθ
r
G◦
kk
(
eiθ
r
))
dθ.
The rest of the proof coincides with the proof of Theorem 4.4.
7. Appendix
Proposition 7.1. (i) For the ensemble (1.3) for |z| < 1 we have
in the case 0 ≡ f(z) = ∆S,T (z),
〈G(z1)SG(z2)〉kk =
f
′
T (0)
1− z1z2f
′
S(0)f ′T (0)
S∗kk + O(n−1); (7.1)
in the case 0 ≡ f(z) = fS(z) = ∆T (z),
〈G(z1)SG(z2)〉kk = f
′
T (0)S∗kk + O(n−1); (7.2)
in the case 0 ≡ f(z) = fT (z) = ∆S(z),
〈G(z1)SG(z2)〉kk = O(n−1), (7.3)
and in the case f(z) 6= 0 for |z| < 1,
〈G(z1)SG(z2)〉kk =
δ∆SδρS
δzδ(zf)
SG̃S(z1)G̃S(z2) + O(n−1). (7.4)
(ii) For the ensemble (1.4) we have in the case 0 ≡ f(z) for |z| < 1,
E {G(z1)V ∗SV G(z2)}kk = O(n−1),
and in the case f(z) 6= 0 for |z| < 1,
E {G(z1)V ∗SV G(z2)}kk =
δ∆S
δz
+ O(n−1).
P r o o f. (i) Taking in Proposition 3.1 with Φ = (G(z1)SG(z2))ac, we obtain
〈(G(z1)S [X,U∗TU ] G(z1)SG(z2))ac〉
+ 〈(G(z1)SG(z2)S [X, U∗TU ] G(z2))ac〉 = 0.
Then taking X = E(a,b) and applying the operation n−1
n∑
a=1
to the result, we get
the matrix equality
〈δn,S(z1)U∗TUG(z1)SG(z2)〉 − 〈(1 + z1gn(z1))G(z1)SG(z2)〉
+
〈(
1
n
TrG(z1)SG(z2)S
)
U∗TUG(z2)
〉
−
〈(
1
n
TrG(z1)SG(z2)SU∗TU
)
G(z2)
〉
= 0. (7.5)
480 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
Regrouping the terms by using the resolvent identity and centralized values, we
obtain
− 1 + z1fn(z1)
z1
(S − ρn,S(z1)I) 〈U∗TUG(z1)SG(z2)〉
+
〈
1
n
TrG(z1)SG(z2)S
〉
〈U∗TUG(z2)〉
= −1 + z1fn(z1)
z1
S 〈G(z2)〉+
δ(z∆S)
δz
〈G(z2)〉+ O(n−1).
Then taking the inverse G̃S(z) and regrouping the terms, we have
〈U∗TUG(z1)SG(z2)〉
= 〈G(z2)〉+
z1
1 + z1fn(z1)
(〈
1
n
TrG(z1)SG(z2)S
〉
+
δ∆S
δz
ρn,S(z2)
)
G̃S(z1)G̃S(z2) + O(n−1). (7.6)
After taking the operation 1
nTr · over (7.6), we obtain the following relation:
〈
1
n
TrU∗TUG(z1)SG(z2)
〉
(7.7)
− z1
1 + z1fn(z1)
1
n
Tr G̃S(z1)G̃S(z2)
〈
1
n
TrG(z1)SG(z2)S
〉
= fn(z)− z1ρn,S(z2)
1 + z1fn(z1)
δ∆S
δz
1
n
Tr G̃S(z1)G̃S(z2) + O(n−1).
Note that in the case f(z) ≡ 0, we have 1
nTr G̃S(z1)G̃S(z2) = 1
nTrG2
S(0) = f
′
S(0).
On other hand, let us consider the modified resolvent
Ĝ = UGU∗ = (USU∗T − zI)−1.
Due to the trace property, we have the following identities:
1
n
TrTĜ(z1)T−1Ĝ(z2) =
1
z1
(
1
n
TrU∗TUG(z1)SG(z2)− gn(z2)
)
,
〈
1
n
TrUSU∗Ĝ(z1)T−1Ĝ(z2)
〉
=
1
z1
(〈
1
n
TrG(z1)SG(z2)S
〉
− 1
n
TrT−1G̃T (z2)
)
+O(n−2).
Then, using the identities above and the left invariant analog of Proposition 3.1
with Φ = (Ĝ(z1)T−1Ĝ(z2))ac and the procedure similar to that used above, we
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 481
V. Vasilchuk
finally obtain
− z2
1 + z2fn(z2)
1
n
Tr G̃T (z1)G̃T (z2)
〈
1
n
TrU∗TUG(z1)SG(z2)
〉
+
〈
1
n
TrG(z1)SG(z2)S
〉
=
1
1 + z2fn(z2)
(
1− ρT (z1)z2
δ∆S
δz
)
1
n
Tr G̃T (z1)G̃T (z2) + O(n−1). (7.8)
Thus, for the pair
(〈
1
nTrU∗TUG(z1)SG(z2)
〉
,
〈
1
nTrG(z1)SG(z2)S
〉)
, we have the
following linear system in the case 0 ≡ f(z) = ∆S,T (z):
〈
1
n
TrU∗TUG(z1)SG(z2)
〉
− z1f
′
S(0)
〈
1
n
TrG(z1)SG(z2)S
〉
=O(n−1)
−z2f
′
T (0)
〈
1
n
TrU∗TUG(z1)SG(z2)
〉
+
〈
1
n
TrG(z1)SG(z2)S
〉
= f
′
T (0) + O(n−1).
Solving this system and substituting the solution
〈
1
n
TrG(z1)SG(z2)S
〉
=
f
′
T (0)
1− z1z2f
′
S(0)f ′T (0)
+ O(n−1)
into (7.6), we have for f(z) ≡ 0 in all cases
〈U∗TUG(z1)SG(z2)〉=〈G(z2)〉+z1
〈
1
n
TrG(z1)SG(z2)S
〉
G̃S(z1)G̃S(z2)+O(n−1),
and hence for 0 ≡ f(z) = ∆S,T (z) we have
〈U∗TUG(z1)SG(z2)〉 = 〈G(z2)〉+ z1
f
′
T (0)
1− z1z2f
′
S(0)f ′T (0)
G̃S(z1)G̃S(z2) + O(n−1).
Then, using the consequence of the resolvent identity
〈G(z1)SG(z2)〉 =
1
z1
(S 〈U∗TUG(z1)SG(z2)〉 − S 〈G(z2)〉) , (7.9)
we obtain (7.1) for 0 ≡ f(z) = ∆S,T (z).
Analogously, for 0 ≡ f(z) = fS(z) = ∆T (z), we obtain the system
〈
1
n
TrU∗TUG(z1)SG(z2)
〉
= O(n−1),
〈
1
n
TrG(z1)SG(z2)S
〉
= f
′
T (0) + O(n−1),
482 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
On the Fluctuations of Entries of Matrices...
and hence we get (7.2). For the case 0 ≡ f(z) = fT (z) = ∆S(z), we obtain the
system
〈
1
n
TrU∗TUG(z1)SG(z2)
〉
= O(n−1),
〈
1
n
TrG(z1)SG(z2)S
〉
= O(n−1)
and then (7.9).
Now we suppose f(z) 6= 0 and multiply the relation (7.5) by S and take 1
nTr ·
over the result. Thus we obtain
〈
1
n
TrG(z1)SG(z2)S
〉
=
δ∆Sδ(z∆S)
δzδ(zf)
+ O(n−1).
Substituting this expression into (7.6) and then using (7.9), we obtain (7.4).
(ii) The first relation in (ii) is obtained from (7.1)–(7.3) by substituting V ∗SV
instead of S and taking E {·} since either 1
nTrS∗ = 0 or f
′
T (z) ≡ 0. The second
relation in (ii) is obtained from (7.4) in the same way.
Acknowledgement. This work is supported by the Franco-Ukrainian grant
Dnipro 2013–2014. The author is thankful to Prof. L. Pastur and Prof.
M. Shcherbina for helpful discussions.
References
[1] L. Pastur and V. Vasilchuk, On the Law of Addition of Random Matrices. — Comm.
Math. Phys. 47 (2000), 1–30.
[2] L. Pastur and V. Vasilchuk, On the Law of Addition of Random Matrices: Covari-
ance and the Central Limit Theorem for Traces of Resolvent. — CRM Proc. Lect.
Notes 42 (2007), 399–416.
[3] L. Pastur and M. Shcherbina, Eigenvalue Distribution of Random Matrices. Math.
Surv. Monogr. 171, RI, AMS, 2011.
[4] M. Shcherbina, Central Limit Theorem for Linear Eigenvalue Statistics of the
Wigner and Sample Covariance Random Matrices. — J. Math. Phys., Anal., Geom.
7 (2011), No. 2, 176–192.
[5] G. Chistyakov and F. Götze, The Arithmetic of Distributions in Free Probability
Theory. mathOA/0508245.
[6] V. Vasilchuk, On the Law of Multiplication of Random Matrices. — Math. Phys.
Anal. Geom. 4 (2001), 1–36.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4 483
V. Vasilchuk
[7] A. Lytova and L. Pastur, Fluctuations of Matrix Elements of Regular Functions of
Gaussian Random Matrices. — J. Stat. Phys. 134 (2009), 147–159.
[8] A. Lytova, On Non-Gaussian Limiting Laws for Certain Statistics of Wigner Ma-
trices. — J. Math. Phys., Anal., Geom. 9 (2013), No. 4, 536–581.
[9] S. O’Rourke, D. Renfrew, and A. Soshnikov, On Fluctuations of Matrix Entries of
Regular Functions of Wigner Matrices with Non-Identicaly Distributed Entries. —
J. Theor. Probab. 26 (2013), No. 3, 750–780.
484 Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 4
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