V. L. Girko
The Circular Law. Twenty years later. Part III
In this paper we apply the REFORM method[3, 4] to the deduction of the system of canonical
equations for normalized spectral functions of the matrices An
+Bn
Un
(
ε
)Cn
, where An
,Bn
and
Cn
are nonrandom matrices, and Un
(ε) is random matrix from class C12
[26] or from the following
class C13
of distributions of random matrices:
where are independent random real matrices
whose entries are independent for every n,
and for a certain δ > 0 the Lyapunov condition is fulfilled: This problem has been considered in some
publications for matrices An
+ Un
, where Unitary matrix Un
from the class C1
[26] is distributed by Haar measure, at the “ad hoc” level, without a strong proof, on the basis of heuristic calculations.
Therefore, the behavior of limit n.s.f. of the sum of a random Unitary matrix Un
and a nonrandom
matrix An
has not been discovered. Many conclusions of various kinds have been presented in the
literature (see [13–27]) for random matrices An
+ Ξ
n
Bn
, where Ξ
n
is the random matrix with independent
entries, concerning, for example, the effectiveness of the REFORM method and the role of
the martingale difference representation for the resolvent of random matrices.
We give the formulation of the Circular Law for random matrices An
+ Bn
Un
(ε)Cn
, where An
,Bn
and Cn
are nonrandom matrices, and Un
(ε) is from the so called Class C12
of random
Unitary matrices (see[26]). In this case the Circular law means that the support of accompanying
spectral density p(x, y) of eigenvalues ofAn
+ Bn
Un
(ε)Cn
looks like several drops of mercury on
a table, and inside of some drops it is possible that some dry circles appear. We call this support of
limit density p(x, y) the Mercury support. If the distances between the centers of these drops are large
enough we have several separate almost circular drops as for corresponding description of limit spectral
density for the Hermitian random matrices [28, 29]. According to the distances between the centers of
these drops, these drops might not touch each other or can merge creating fanciful shapes represented
at the end of the paper, Figures 1–16. The analogy with the Circular Law is the following: for a simple
Unitary matrix Un
from class C1
all its eigenvalues are distributed asymptotically uniformly on the
circumference and, evidently, this limit distribution does not coincide with the uniform distribution on
the circle. But if we will add to Unitary matrix Un
any diagonal nonrandom matrix An
= (δijaj
) then if the distances between diagonal entries aj
are large enough we have almost the same picture
of limit distribution of eigenvalues of Un
+ An
if Unitary matrix Un
would be equal to the random matrix Ξ
n
with random independent entries satisfying the Global Circular Law (see Part I of this
paper). The rough explanation of this phenomenon is the following: the expected scalar products of
the vector row or vector column of matrix Ξ
n
have the order n
−1. Therefore, the matrix Ξ
n
looks like orthogonal random matrix. In this paper we use for matrices An
+Bn
Un
(ε)Cn
the triply regularized
V -transform[2, 5],(VICTORIA-transform of random matrix which is the abbreviation of the following
words: Very Important Computational Transformation Of Randomly Independent Arrays)
where α > 0, ε ≠= 0, ? ≠= 0 are regularization parameters , τ = t + is is a complex number and µn
{x,Gn
(t, s, ?, ε)} is the normalized spectral function of G-matrix:
Such a V -transformation was used for the first time in 1982–1985 in [2,3,5]. Canonical equations for
the limit distribution function of normalized spectral functions of some matrices An
+ BnUn
and
the following V-domain
Gδ, δ > 0(see [27, pp.142, 173]) of the pseudodistribution of eigenvalues of
matrices An
+ BnU
n
are found:
where are eigenvalues of the
matrix.
Random Operators and Stochastic Equations, Walter de Gruyter
Print ISSN: 0926-6364
Volume: 13, 01/2005
Pages: 53 - 109
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