Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach
We study an eigenvalue distribution of the adjacency matrix A(N,p,α) of the weighted random bipartite graphs
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irk-123456789-1405482018-07-11T01:23:05Z Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach Vengerovsky, V. We study an eigenvalue distribution of the adjacency matrix A(N,p,α) of the weighted random bipartite graphs Исследуется распределение собственных значений матрицы смежности случайного двудольного графа. 2016 Article Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach / V. Vengerovsky // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 1. — С. 78-93. — Бібліогр.: 17 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag12.01.078 Mathematics Subject Classification 2000: 15B52. http://dspace.nbuv.gov.ua/handle/123456789/140548 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study an eigenvalue distribution of the adjacency matrix A(N,p,α) of the weighted random bipartite graphs |
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Vengerovsky, V. |
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Vengerovsky, V. Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach Журнал математической физики, анализа, геометрии |
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Vengerovsky, V. |
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Vengerovsky, V. |
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Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach |
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Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach |
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Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach |
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Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach |
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Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach |
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eigenvalue distribution of bipartite large weighted random graphs. resolvent approach |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Eigenvalue Distribution of Bipartite Large Weighted Random Graphs. Resolvent Approach / V. Vengerovsky // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 1. — С. 78-93. — Бібліогр.: 17 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT vengerovskyv eigenvaluedistributionofbipartitelargeweightedrandomgraphsresolventapproach |
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2025-07-10T10:42:19Z |
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2025-07-10T10:42:19Z |
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1837256295014465536 |
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Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 1, pp. 78–93
Eigenvalue Distribution of Bipartite Large Weighted
Random Graphs. Resolvent Approach
V. Vengerovsky
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Nauki Ave., Kharkiv, 61103, Ukraine
E-mail: vengerovsky@ilt.kharkov.ua
Received January 26, 2015, revised June 16, 2015,
published online December 4, 2015
We study an eigenvalue distribution of the adjacency matrix A(N,p,α)
of the weighted random bipartite graphs Γ = ΓN,p,α. We assume that the
graphs have N vertices, the ratio of parts is
α
1− α
, and the average number
of edges attached to one vertex is αp for the first part and (1− α)p for the
second part of vertices. To each edge of the graph eij , we assign the weight
given by a random variable aij with the finite second moment.
We consider the resolventsG(N,p,α)(z) of A(N,p,α) and study the functions
f1,N (u, z) =
1
[αN ]
[αN ]∑
k=1
e−ua
2
kG
(N,p,α)
kk (z)
and
f2,N (u, z) =
1
N − [αN ]
N∑
k=[αN ]+1
e−ua
2
kG
(N,p,α)
kk (z)
in the limit N →∞. We derive a closed system of equations that uniquely
determine the limiting functions f1(u, z) and f2(u, z). This system of equa-
tions allows us to prove the existence of the limiting measure σp,α. The weak
convergence in probability of the normalized eigenvalue counting measures
is proved.
Key words: sparse random matrices, bipartite graphs, normalized eigen-
value counting measure.
Mathematics Subject Classification 2010: 15B52.
c© V. Vengerovsky, 2016
Eigenvalue Distribution of Bipartite Large Weighted Random Graphs
1. Introduction
Random graphs appear in different branches of mathematics and physics (see
monographs [5, 7] and references therein). It is well known that they are closely
connected with the theory of random matrices, since there is a one to one map
between the graphs with N vertices and their adjacency matrices (recall that by
definition the entries aij of the adjacency matrix are 1 if the vertices i and j are
connected, and aij = 0 otherwise). The spectrum of the graph is the set of n
eigenvalues of the adjacency matrix.
The model of a random graph, introduced by P. Erdos (see, e.g., [7] and
references therein), is a simple but very important and popular sample of random
graphs. In this model, usually called the G(N, pN ) random graph, each pair of
N graph vertices is connected by the edge with probability pN , independently
from all other edges of the graph. The corresponding ensemble of random N ×N
adjacency matrices M can be represented as M = {mij}Ni,j=1 with mii = 0 and
i.i.d.
mij ==
{
1, with probability pN ,
0, with probability 1− pN .
(1.1)
The ensemble of adjacency matrices (1.1) is a particular case of the random
matrix ensemble. The eigenvalue distribution of different ensembles of N × N
random matrices, as N → ∞, was intensively studied during a half of century
(see [10] and references therein) since the pioneer work of Wigner [17]. One of
the questions of primary interest is the asymptotic behavior, as N → ∞, of
the so-called Normalized Eigenvalue Counting Measure (NCEM) defined by the
formula
Nn(λ)=
]{j : λ(N)
j <λ}
N
,
where {λ(N)
j }Nj=1 are the eigenvalues of the matrix under consideration. Ensemble
(1.1) plays an important role in the random matrix theory not only because of
its links with random graphs, but also because, varying pN from O(1/N) to O(1)
(as N → ∞), one can interpolate between the matrices of Wigner type, which
have almost all entries nonzero, and the so-called sparse random matrices with
pN = O(1/N). The intermediate case, when 1 << pN << 1, is also interesting,
and it is studied both in physics and mathematics. In particular, in [9], the
limiting eigenvalue distribution of ensemble (1.1) was found, and it appears that
the distribution is the same as that for the classical Wigner model [17]. The
most interesting sparse regime (pN = O(1/N)) for ensembles (1.1) was studied
intensively first in physics literature, where the convergence of NCEM to some non
random limiting measure was established on the physical level of rigor (see [12,
13]). Then in the papers [2, 3] the same results were obtained on the mathematical
level of rigor by using the so-called moment approach based on the convergence
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 79
V. Vengerovsky
of the NCEM moments to some non random limits. In [8], the similar results for
NCEM were obtained for the case of the so-called weighted random graphs, which
differ from (1.1) by some random independent from {mij}1≤i<j≤N multipliers
{aij}1≤i<j≤N , usually called weights. The spectrum for sparse random graphs
with subgaussian weigths coincides with the whole real line. Also, in this case the
tails decay as exp{−cx2log(x)} as x → ∞. The ensembles of weighted random
graphs are now especially interesting because of their links with the so-called
ensembles with blowing moments and also ensembles of matrices with heavy tales
[4]. The next step in studying the eigenvalue distribution of large random graphs
(the proof of Central Limit Theorem for linear eigenvalue statistics) was done in
[14] for the sparse regime and in [15] for the diluted regime.
In the present paper, we consider the bipartite analogs of large sparse random
graphs. The moment approach for this ensemble was developed in [16], where it
was shown that the moments of the corresponding NCEM have the limits which
satisfy the system of recursive relations. But this approach requires that rather
restrictive conditions be imposed on the moments of the weights {aij}1≤i<j≤N .
We apply the resolvent approach, developed in [8] for the case of standard random
weighted graphs, to the case of bipartite graphs. This allows us to optimize the
conditions imposed on the moments of {aij}1≤i<j≤N .
2. Main Results
Let us introduce the randomly weighted adjacency matrix of random bipartite
graphs. Let Ξ = {aij , i≤j, i, j∈N} be the set of jointly independent identically
distributed (i.i.d.) random variables determined on the same probability space.
We set aji=aij for i≤j .
Given 0< p≤N , let us define the family D(p)
N ={d(N,p)
ij , i≤ j, i, j ∈ 1, N} of
jointly independent random variables
d
(N,p)
ij =
{
1, with probability p/N,
0, with probability 1− p/N. (2.1)
We set dji = dij and assume that D(p)
N is independent from Ξ.
Let α ∈ (0, 1), define Iα,N = 1, [α ·N ], where [·] is an integer part of the
number. Now one can consider the real symmetric N ×N matrix A(N,p,α)(ω):
[
A(N,p,α)
]
ij
=
{
aijd
(N,p)
ij , if (i ∈ Iα,N ∧ j /∈ Iα,N ) ∨ (i /∈ Iα,N ∧ j ∈ Iα,N ),
0, otherwise
(2.2)
that has N real eigenvalues λ(N,p,α)
1 ≤λ(N,p,α)
2 ≤ . . .≤ λ
(N,p,α)
N .
80 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Eigenvalue Distribution of Bipartite Large Weighted Random Graphs
The normalized eigenvalue counting function of A(N,p,α) is determined by the
formula
σ
(
λ;A(N,p,α)
)
=
]{j : λ(N,p,α)
j <λ}
N
.
We denote by F the class of functions which are analytic with respect to z :
<z > 0 and for any fixed z : <z > 0 possessing the norm
||f(u, z)|| = sup
u>0
|f(u, z)|√
1 + u
. (2.3)
Theorem 1. Assume that the µ(a) = E{θ(a − ai,j)} probability distribution
of ai,j possesses the property ∫
a4dµ(a) = X4 <∞. (2.4)
(We denote by Xi the i-th absolute moment of a, i.e., Xi =
∫
|a|idµ(a).) Then
the measure σ
(
λ;A(N,p,α)
)
converges weakly in probability to nonrandom measure
σp,α,
σ
(
· ;A(N,p,α)
)
w,P→ σp,α, N →∞. (2.5)
The Stieltjes transform gσp,α of the limiting measure σp,α can be found as follows:
gσp,α(z) = −ih(iz), (2.6)
h(z) : C+ → C+, h(z) = α · h1(z) + (1− α) · h2(z), (2.7)
hi(z) = −X−1
2
∂
∂u
fi(u, z)
∣∣∣∣
u=0
, i = 1, 2, (2.8)
where a pair f1(u, z) and f2(u, z) is a unique solution of the following system of
functional equations in the class F :{
f1(u, z) = L(f2, µ, 1− α)
f2(u, z) = L(f1, µ, α)
, (2.9)
where
L(f, µ, α) = 1− u1/2e−αp
∫
|a|dµ(a)
∞∫
0
dv
J1(2|a|
√
uv)√
v
exp{−zv + αpf(v, z)},
(2.10)
J1(ζ) is the Bessel function
J1(ζ) =
ζ
2
∞∑
k=0
(−ζ2/4)k
k!(k + 1)!
. (2.11)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 81
V. Vengerovsky
Proposition 1. Condition (2.4) in Theorem 1 can be replaced by∫
a2dµ(a) = X2 <∞
via the truncation procedure.
The proof of Proposition 1 is given in Section 4..
Theorem 1 is a corollary of Theorem 2.
Theorem 2. Let the distribution of aj,k satisfy condition (2.4). Then
(i) the variance of gN,p,α(z) vanishes in the limit N →∞:
E{|gN,p,α(z)− E{gN,p,α(z)}|2} ≤ C(z, p, α,X2)
N1/2
, (2.12)
(ii) there exists the limiting probability measure σp,α such that
gσp,α(z) = lim
N→∞
E{gN,p,α(z)} = −ih(iz), (2.13)
for an arbitrary compact in C+, the convergence is uniform, and the function
h(z) : C+ → C+ can be expressed in terms of the pair of functions f1(u, z) and
f2(u, z) (see (2.7)–(2.8)), which is a unique solution of the system of functional
equations (2.9) in the class F .
3. Proof of Theorem 1
For any z: <z > 0, consider the functions f1,N (u, z) : R+ → C, f2,N (u, z) :
R+ → C:
f1,N (u, z) =
1
[αN ]
[αN ]∑
k=1
e−ua
2
kG
(N,p,α)
kk (z), G(N,p,α)(z) = (z − iA(N,p,α))−1, (3.1)
f2,N (u, z) =
1
N − [αN ]
N∑
k=[αN ]+1
e−ua
2
kG
(N,p,α)
kk (z),
where {ai}∞i=1 is a family of i.i.d. random variables which do not depend on
{ai,j}∞i<j and have the same probability distribution as a1,2. It is easy to see that
G
(N,p,α)
NN (z) can be represented in the form
G
(N,p)
NN (z) =
(
z − iA(N,p,α)
NN +
N−1∑
j,k=1
G̃
(N−1,p,α)
jk A
(N,p,α)
Nj A
(N,p,α)
Nk
)−1
, (3.2)
82 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Eigenvalue Distribution of Bipartite Large Weighted Random Graphs
where the matrix {G̃(N−1,p,α)
ij (z)}Ni,j=2 is a resolvent of the matrix iÃ(N−1,p,α),
which can be obtained from A(N,p,α) by deleting the last column and the last
row. Let us use the formula (see [1]):
e−ua
2R = 1− u1/2|a|
∞∫
0
dv
J1(2|a|
√
uv)√
v
exp{−R−1v}, (3.3)
which is valid for any u ≥ 0, <R > 0. Then, on the basis of (3.2), we get
exp{−ua2
NG
(N,p,α)
NN } = 1− u1/2|aN |
∞∫
0
dv
J1(2|aN |
√
uv)√
v
exp{−zv − v
[αN ]∑
j,k=1
G̃
(N−1,p,α)
ij A
(N,p,α)
Ni A
(N,p,α)
Nj }.
(3.4)
Denote
RN (z) =
[αN ]∑
j,k=1,j 6=k
G̃
(N−1,p)
jk A
(N,p,α)
Nj A
(N,p,α)
Nk . (3.5)
Proposition 2.
E{|R1(z)|2} ≤ 2
p2X2
2
N |<z|2
+
p4X4
1
N2|<z|2
+ 4
p3X2
1X2
N2|<z|2
+ 6
p4X4
1
N |<z|2
+ 8
p3X2
1X2
N |<z|2
, (3.6)
where
Xk =
∫
|a|kdµ(a).
P r o o f.
E{|R1(z)|2} =
[αN ]∑
6={j1,j2,k1,k2}
E
{
G̃
(N−1,p)
j1k1
G̃
(N−1,p)
j2k1
A
(N,p)
Nj1
A
(N,p)
Nj2
A
(N,p)
Nk1
A
(N,p)
Nk2
}
+ 4
[αN ]∑
6={j,k1,k2}
E
{
G̃
(N−1,p)
jk1
G̃
(N−1,p)
jk1
|A(N,p)
Nj |
2A
(N,p)
Nk1
A
(N,p)
Nk2
}
+ 2
[αN ]∑
j 6=k
E
{
G̃
(N−1,p)
jk G̃
(N−1,p)
jk |A(N,p)
Nj |
2|A(N,p)
Nk |
2
}
= H1 + 4H2 + 2H3.
(3.7)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 83
V. Vengerovsky
Averaging with respect to {A(N,p)
N,i }
N−1
i=1 and using the fact that {G̃(N−1,p)
ij (z)}N−1
i,j=1
do not depend on A
(N,p)
N,i , we obtain
H1 ≤ X4
1
p2
N2
E
{∣∣∣∣N−1
[αN ]∑
j,k
Ĝjk
∣∣∣∣2}+ 6
p4X4
1
N |<z|2
≤ p2X4
1
N2|<z|2
+ 6
p4X4
1
N |<z|2
,
H2 ≤ X2
1X2
p
N3
[αN ]∑
k1 6=k2
E
{
[ĜĜ∗]k1k2
}
+ 2
p3X2
1X2
N |<z|2
≤ pX2
1X2
N2|<z|2
+ 2
p3X2
1X2
N |<z|2
,
H3 ≤ X2
2
N2
[αN ]∑
k
E
{
[ĜĜ∗]kk
}
≤ X2
2
N |<z|2
.
Besides, since evidently that
<
{∑
G̃
(N−1,p)
ij A
(N,p,α)
Ni A
(N,p,α)
Nj
}
≥ 0, <
{∑
G̃
(N−1,p,α)
jj |A(N,p,α)
jj |2
}
≥ 0,
the inequality
|e−z1 − e−z2 | ≤ |z1 − z2| (<z1,<z2 ≥ 0) (3.8)
and (3.4) imply
E exp{−ua2
NG
(N,p,α)
NN } = 1− u1/2E|aN |
∞∫
0
dv
J1(2|aN |
√
uv)√
v
×E2 exp{−zv − v
[αN ]∑
i=1
G̃
(N−1,p,α)
ii |A(N,p,α)
Nj |2}+ r̃N (u),
(3.9)
where E2 denotes the averaging over {aij}i,j and
{
d
(N,p)
ij
}
i,j
. Remainder r̃N (u)
obeys the following condition:
|r̃N (u)| ≤ E|R1|u1/2E|aN |
∞∫
0
dv|J1(2|a1|
√
uv)√
v
e−zv| ≤ C|R1|u1/2E|aN ||<z|−1/2.
In the last inequality we use the estimate for the Bessel function
|J1(u)| ≤ 1. (3.10)
Here and below we denote by C some constants (different in different formu-
las), which do not depend on N, z, p, α. Taking into account (3.6), we get
|r̃N (u)| ≤ C(p,X2)u1/2
√
N |<z|5/2
. (3.11)
84 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Eigenvalue Distribution of Bipartite Large Weighted Random Graphs
Since G̃(N−1,p)(z) does not depend on
{
d
(N,p)
N,j
}N
j=1
, we obtain
E2 exp{−v
[αN ]∑
i=1
G̃
(N−1,p,α)
ii |A(N,p,α)
Nj |2} = E
{ [αN ]∏
j=1
(
(1− p
N
) +
p
N
e−va
2
NjG̃
(N−1,p)
jj
)}
= e−αpE
{
exp{αpf̃1,N−1(v, z)}
}
+RN (v),
(3.12)
where
f̃1,N−1(u, z) =
1
[αN ]
[αN ]∑
k=1
e−va
2
NjG̃
(N,p,α)
kk (z),
and RN satisfies the estimate
|RN (v)| ≤ Cp2
N
.
Using (3.8), we get∣∣∣E exp
{
αpf̃1,N−1(v, z)
}
− exp
{
αpEf̃1,N−1(v, z)
}∣∣∣ ≤
≤ αpeαpE
∣∣∣f̃1,N−1(v, z)− Ef̃1,N−1(v, z)
∣∣∣ . (3.13)
Further considerations are based on Lemma 1.
Lemma 1. Fix α ∈ (0, 1). Let A(n) be a real symmetric n × n matrix such
that
A
(n)
ij =
{
aij , if (i ∈ Iα,n ∧ j /∈ Iα,n) ∨ (i /∈ Iα,n ∧ j ∈ Iα,n),
0, otherwise
(3.14)
where {aij}1≤i≤j is a family of jointly independent identically distributed random
variables that obey the following conditions:
E|aij | ≤
C
n
, Ea2
ij ≤
C
n
. (3.15)
For z: <z > 0, consider
R = (z − iA)−1, Fn(z) = [αn]−1
[α ·n]∑
j=1
ϕj(Rjj), (3.16)
where random functions ϕj satisfy the condition
|ϕ′j(ζ)| ≤ C3 αj . (3.17)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 85
V. Vengerovsky
where {αj} is a set of jointly independent identically distributed random variables
also independent of {aij} such that E
{
α2
1
}
<∞. Then
V arFn(z) ≤ 4(1− α)C2
3
αn|<z|4
E
{
α2
1
} (
nE|a12|+ (nEa2
12)2
)
. (3.18)
The proof of Lemma 1 is given in Section 4.
R e m a r k 1. Lemma 1 is still valid for Fn(z) = n−1
∑n
j=1 ϕj(Rjj) with
changed constants.
Lemma 1 for ϕ(ζ) = exp {−va2
Njζ}, αj = va2
Nj , C3 = 1, n = N − 1 implies
E
∣∣∣f̃1,N−1(v, z)− Ef̃1,N−1(v, z)
∣∣∣2 ≤ C̃2(X4, p)v2
αN |<z|4
. (3.19)
Relations (3.13) and (3.19) yield∣∣∣E exp
{
αpf̃1,N−1(v, z)
}
− exp
{
αpEf̃1,N−1(v, z)
}∣∣∣ ≤ αpeαp C̃(X4, p)v
α1/2N1/2 |<z|2
.
(3.20)
Combining (3.4)–(3.20), we get
Ef2,N (u, z) = 1− u1/2e−αp
∞∫
0
dve−zv
J1(2
√
uv)√
v
eαpEf̃1,N−1(v,z) + r(u), (3.21)
r(u) ≤
C̃(Ea4
Nj , p)u
1/2
N1/2 |<z|7/2
.
In order to get the closed system of equations, we have to replace f̃1,N−1 by f1,N .
For this purpose we use the next bound on their difference∣∣∣Ef1,N (v, z)− Ef̃1,N−1(v, z)
∣∣∣ ≤ C̃(X4, p)v
αN1/2 |<z|2
. (3.22)
Indeed, using Lemma 1 for ϕ(ζ) = exp {−va2
jζ}, αj = va2
j , C3 = 1, n = N , we
get
E |f1,N (v, z)− Ef1,N (v, z)|2 ≤ C̃2(X4, p)v2
αN |<z|4
. (3.23)
Combining (3.23), (3.19) and (4.6) for ϕ(ζ) = exp {−va2
Njζ}, αj = va2
Nj , C3 = 1,
n = N , we obtain (3.22).
86 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Eigenvalue Distribution of Bipartite Large Weighted Random Graphs
The inequalities (3.22), (3.8) and (3.21) imply
Ef2,N (u, z) = 1− u1/2e−αp
∞∫
0
dve−zv
J1(2
√
uv)√
v
eαpEf1,N (v,z) + r(u), (3.24)
r(u) ≤ C̃(X4, p)u1/2
N1/2 |<z|7/2
.
Let us consider the Banach space H of all the functions h : R+ → C which
possess the norm (2.3). The space H2 possesses the norm ||(h1, h2)||H2 =
max {||h1||H, ||h2||H} .
Define the operator Fz : H2 → H2 of the form
Fz(ϕ1, ϕ2) = (ψ1, ψ2), ψ1(u) = L(f2, µ, 1− α), ψ2(u) = L(f1, µ, α). (3.25)
Let us denote by B0,2 = {h ∈ H2, ‖h‖H2 ≤ 2} the ball of radius 2 centered in the
origin. Then for any ϕ1, ϕ2 : ||ϕ1|| ≤ 2, ‖ϕ2‖ ≤ 2,
‖Fz(ϕ1)− Fz(ϕ2)‖ ≤ X1pe
2p+ 2p2
|<z|
(
2
|<z|
+
4
|<z|1/2
)
‖ϕ1 − ϕ2‖. (3.26)
Indeed, inequalities (3.8), (3.10) imply
‖Fz(ϕ1)− Fz(ϕ2)‖H2 ≤ X1p‖ϕ1 − ϕ2‖
∞∫
0
dv(1 + v1/2)e−|<z|v+2p(1+v1/2)
√
v
. (3.27)
Using the trivial inequality
2pv1/2 − v|<z|/2 ≤ 2
p2
|<z|
, (3.28)
we get (3.26).
It is easy to see that ‖Fz(0)‖ obeys the inequality
‖Fz(0)‖H2 ≤ sup
u>0
1 + 2X1|<z|−1/2u1/2
1 + u1/2
≤ 1 + 2X1|<z|−1/2. (3.29)
Thus there is M > 0 such that
‖Fz(ϕ1)− Fz(ϕ2)‖ < 1/4‖ϕ1 − ϕ2‖, ‖Fz(0)‖ ≤ 5
4
, z ∈ L(M), (3.30)
where L(M) = {z : |<z| > M}.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 87
V. Vengerovsky
Therefore, Fz : B0,2 → B0,2, and Fz|B0,2 is a contraction mapping for all
z ∈ L(M). Hence there exists the unique fixed point f(u, z) which is a solution
of (2.9). Since |Ef1,N (u, z)| ≤ 1, |Ef2,N (u, z)| ≤ 1, EfN (u, z) ∈ B0,2, then the
estimates (3.30) imply
EfN (u, z) = Fz(EfN (u, z)) + rN (u, z) = . . . = f(u, z) + r∞N (u, z), (3.31)
where
‖r∞N (u, z)‖ ≤ ‖rN (u, z)‖
∞∑
k=0
1
4k
≤ 4C(p,M)
3N1/2
, z ∈ L(M). (3.32)
Hence, EfN (u, z) ⇒ f(u, z), z ∈ L(M). Fix u. Since Ef1,N (u, z), Ef2,N (u, z)
are analytic and uniformly bounded for arbitrary Πε,a = {z : ε ≤ <z ≤ 2M, |=z| ≤
a}, by the Arzela theorem we can choose a subsequence {Nk}∞k=1 such that
Ef1,Nk(u, z) ⇒ f̃
(a,ε)
1 (u, z), Ef2,Nk(u, z) ⇒ f̃
(a,ε)
2 (u, z) in Πε,a. Then f
(a,ε)
1 (u, z),
f
(a,ε)
2 (u, z) are analytic in Πε,a. But
f
(a,ε)
2 (u, z) = f2(u, z), f (a,ε)
1 (u, z) = f1(u, z), |<z| > M.
The uniqueness theorem of complex analysis and the arbitrariness of choos-
ing the subsequence imply the existence of the analytic extension of f1(u, z)
(f2(u, z) in C+ and the uniform convergence in z fα,N (u, z) ⇒ fα(u, z) for an
arbitrary compact in C+. Thus, if we fixed any z : <z > 0, we obtain that
fα,N (u, z), (α = 1, 2) as a function of u converges pointwise to fα,N (u, z). But
since | ∂∂ufα,N (u, z)| ≤ C and |fα,N (u, z)| ≤ 1, the pointwise convergence im-
ply also the convergence in the norm (2.3). Then, using Lebesgue’s dominated
convergence theorem, we can prove that f(u, z) = Fz(f)(u, z) in C+.
Indeed, using Lemma 1 for ϕ(ζ) = ζ, αj = 1, C3 = 1, n = N , we get (2.12).
Uniform convergence fN (u, z) in u ∈ [0, 1] imply
EgN,p,α = −i(Ea2
1)−1 1
N
N∑
k=1
Ea2
kE{G
(N,p,α)
kk } =
= i(Ea2
1)−1
(
α
∂
∂u
Ef1,N (u, z)
∣∣∣∣
u=0
+ (1− α)
∂
∂u
Ef2,N (u, z)
∣∣∣∣
u=0
)
. (3.33)
The next simple proposition allows us to make a final step.
Proposition 3. Set Ψn(u) =
fn(u)− fn(0)
u
− f ′n(0). Assume that
|Ψn(u)| ≤ ε(u), ε(u)→ 0, as u→ 0. (3.34)
88 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Eigenvalue Distribution of Bipartite Large Weighted Random Graphs
If there exists f(u) = lim
n→∞
fn(u) and the function f is differential at u = 0, then
lim
n→∞
f ′n(0) = f ′(0). (3.35)
P r o o f.∣∣∣f1,n(u, z)− f1,n(0, z)
u
− ∂
∂u
f1,n(u, z)
∣∣∣∣
u=0
∣∣∣
=
∣∣∣∣
1∫
0
(
∂
∂u
f1,n(tu, z)− ∂
∂u
f1,n(u, z)
∣∣∣∣
u=0
)
dt
∣∣∣∣
≤ 1
[αN ]
[αN ]∑
k=1
1∫
0
a2
k
∣∣∣G(N,p,α)
kk
∣∣∣∣∣∣∣e−uta2
kG
(N,p,α)
kk − 1
∣∣∣∣dtdµ(ak)
≤ 1
|<z|
1
[αN ]
[αN ]∑
k=1
∫
|a2|>u−1/2
+
∫
|a2|≤u−1/2
∫ a2
∣∣∣∣e−uta2
kG
(N,p,α)
kk − 1
∣∣∣∣dµ(ak)dt
≤ 1
|<z|
2
∫
|a2|>u−1/2
a2dµ(a) +
1
|<z|
√
u
∫
a2dµ(a)
u→0−−−→ 0. (3.36)
The similar estimate with f2,n is valid too. Hence,
gp,α(z)= lim
N→∞
EgN,p,α(z) = i(Ea2
1)−1
(
α
∂
∂u
f1(u, z)
∣∣∣∣
u=0
+ (1− α)
∂
∂u
f2(u, z)
∣∣∣∣
u=0
)
.
(3.37)
4. Proofs of Auxiliary Statements
P r o o f of Lemma 1. Let us denote by Ek the averaging over {aij}i≤j i ≤ k
(En = E, E0 means the absence of averaging), by n1 = [αn], by n2 = n − n1.
Then
Fn − EFn =
n1−1∑
k=0
(EkFn − Ek+1Fn)⇒ E|Fn − EFn|2
= 2
n1−1∑
j<k
E(EkFn − Ek+1Fn)(EjFn − Ej+1Fn) +
n1−1∑
k=0
E|EkFn − Ek+1Fn|2
=
n1−1∑
k=0
E|EkFn − Ek+1Fn|2. (4.1)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 89
V. Vengerovsky
Here we use the identity E(EkFn − Ek+1Fn)(EjFn − Ej+1Fn) = 0 for j 6= k.
Denote by E(k) the averaging over {akj}nj=k. Let F (k)
n = Fn
∣∣∣∣
{akj=0}n
j=1
. So we
get
E|EkFn − Ek+1Fn|2 = E|Ek(Fn − E(k+1)Fn)|2 ≤ E
∣∣∣Ek (Fn − F (k+1)
n
)∣∣∣2 .
Taking into account the Schwarz inequality, we obtain E
∣∣∣Ek (Fn − F (k)
n
)∣∣∣2 ≤
EEk
∣∣∣(Fn − F (k+1)
n
)∣∣∣2 = E
∣∣∣Fn − F (k+1)
n
∣∣∣2. Due to the symmetry, for all k, we
have
E|EkFn − Ek+1Fn|2 ≤ E
∣∣∣Fn − F (k+1)
n
∣∣∣2 = E
∣∣∣Fn − F (1)
n
∣∣∣2 . (4.2)
In order to estimate E
∣∣∣∣Fn − F (1)
n
∣∣∣∣2, we introduce the matrix A(t) :
Aij(t) =
(
1i∈Iα,N ,j /∈Iα,N + 1i/∈Iα,N ,j∈Iα,N
)
(1i≥1,j≥1aij + t(1− 1i≥1,j≥1)aij) .
Also, we introduce the functions
R(t) = (z − iA(t))−1, Fn(t) = n−1
1
n1∑
j=1
ϕ(Rjj(t)).
Clearly, the equality
Fn − F (1)
n =
1∫
0
d
dt
Fn(t)dt (4.3)
is true. We can estimate
d
dt
F (t) by the following way:
d
dt
Fn(t) =
1
n1
n1∑
j=1
∑
k,l
ξj(t)Rjk(t)A′kl(t)Rlj(t) =
2
n1
∑
j∈Iα,n,k,l
ξj(t)Rj1(t)a1k(t)Rkj(t) = 2H,
where
ξj(t) =
∂
∂Rjj
ϕ(Rjj(t)),
E
{
|H|2
}
≤ C2
3
4n2
1
E
∑
j,j′∈Iα,n,k,k′
αjα
′
j
(
|Rjk|2 + |R1j |2
)
|a1k|
(∣∣Rj′k′∣∣2 +
∣∣R1j′
∣∣2) |a1k′ |
≤ C2
3
n2
1
E
{
α2
1
}
E
(∑
k
|a1k|2
)
≤ C2
3
n2
1|<z|4
E
{
α2
1
} (
n2E|a12|+ (n2Ea2
12)2
)
. (4.4)
90 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
Eigenvalue Distribution of Bipartite Large Weighted Random Graphs
Here we use the inequality∑
j
|Rsj |2 = [RR∗]ss ≤
1
|<z|2
.
Using (4.3) and (4.4), we obtain
E
∣∣∣Fn − F (1)
n
∣∣∣2 = E
∣∣∣∣∣∣
1∫
0
d
dt
Fn(t)
∣∣∣∣∣∣
2
≤
1∫
0
E
{∣∣∣∣ ddtFn(t)
∣∣∣∣2
}
≤ 4C2
3
n2
1|<z|4
E
{
α2
1
} (
n2E|a12|+ (n2Ea2
12)2
)
. (4.5)
In much the same way the following estimate can be proved:
E
∣∣∣Fn − F (n)
n
∣∣∣2 = E
∣∣∣∣∣∣
1∫
0
d
dt
Fn(t)
∣∣∣∣∣∣
2
≤
1∫
0
E
{∣∣∣∣ ddtFn(t)
∣∣∣∣2
}
≤ 4C2
3
n2
1|<z|4
E
{
α2
1
} (
n2E|a12|+ (n2Ea2
12)2
)
. (4.6)
Combining (4.1), (4.2), (4.5), we obtain (3.18).
P r o o f of Proposition 1. Denote by a(T ) the truncation of a with the
parameter T , i.e.,
a(T )(ω)=
{
a(ω), if a(ω) < T,
T, otherwise
. (4.7)
Here and below the notation with an upper index (T ) means that the function
is defined for the matrix A(T ) by the same way as it was done for A.
Similarly to (3.27) and (3.28), we can obtain that for any ϕ: ||ϕ|| ≤ 2,
‖Fz(ϕ)− F (T )
z (ϕ)‖ ≤ 2e2p+
2p2
|<z|
|<z|
‖ϕ‖
∞∫
T
|a|dµ(a). (4.8)
Combining (4.8) and (3.30), we obtain
∀z ∈ L(M) f (T )(u, z)
‖‖H2−−−−→
T→∞
f(u, z). (4.9)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 91
V. Vengerovsky
Theorem 2 yields
g
σ
(T )
p,α
(z) = lim
N→∞
E{g(T )
N,p,α(z)}, (4.10)
and for an arbitrary compact in C+ the convergence is uniform.
Taking into account the resolvent identity and the Schwarz inequality, we
obtain ∣∣∣Eg
σ
(T )
N,p,α
(z)− EgσN,p,α(z)
∣∣∣ ≤ p
N |=z|2
∞∫
T
a2dµ(a)
1/2
. (4.11)
Combining (4.9)–(4.11), we obtain
gσp,α(z) = lim
N→∞
E{gN,p,α(z)}, (4.12)
and for an arbitrary compact in C+ the convergence is uniform.
(2.12) is still valid because it does not require the existence of X4 (just use
Lemma 1 for ϕ(ζ) = ζ, αj = 1, C3 = 1, n = N).
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