Results 1 
7 of
7
Spectral measure of heavy tailed band and covariance random matrices, preprintarXiv:0811.1587v2 [math.PR
 Mathematical Physics
"... Abstract. We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure ˆµ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix Yσ i j N whose (i, j) entry is σ ( , N N)xij where (xij,1 ≤ i ≤ j < ∞ ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Abstract. We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure ˆµ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix Yσ i j N whose (i, j) entry is σ ( , N N)xij where (xij,1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an αstable law, α ∈ (0, 2), and σ is a deterministic function. For random diagonal DN independent of Yσ N and with appropriate rescaling aN, we prove that ˆµ −1 a N Yσ N +D converges in mean towards a limiting probability N measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. 1.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
Rooted trees and moments of large sparse random matrices
"... In these expository paper we describe the role of the rooted trees as a base for convenient tools in studies of random matrices. Regarding the Wigner ensemble of random matrices, we represent main ingredients of this approach. Also we refine our previous result on the limit of the spectral norm of a ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In these expository paper we describe the role of the rooted trees as a base for convenient tools in studies of random matrices. Regarding the Wigner ensemble of random matrices, we represent main ingredients of this approach. Also we refine our previous result on the limit of the spectral norm of adjacency matrix of large random graphs.
Probability density of determinants of random matrices
 J. Phys. A: Math. Gen
"... Abstract. In this brief note the probability density of a random real, complex and quaternion determinant is rederived using the singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices. 1 Introduction and results. We consider ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. In this brief note the probability density of a random real, complex and quaternion determinant is rederived using the singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices. 1 Introduction and results. We consider n×n matrices whose elements are either real, or complex or quaternions (in what follows, the four components of the quaternions will always be real); the real parameters entering these elements are independent gaussian random variables with mean zero and the same variance. The number of real
Real symmetric random matrices and paths counting
, 2004
"... Exact evaluation of Tr < S p> is here performed for real symmetric matrices S of arbitrary order n, where the entries are independent identically distributed random variables, with an arbitrary probability distribution, up to some integer p. These polynomials provide useful information on the spectr ..."
Abstract
 Add to MetaCart
Exact evaluation of Tr < S p> is here performed for real symmetric matrices S of arbitrary order n, where the entries are independent identically distributed random variables, with an arbitrary probability distribution, up to some integer p. These polynomials provide useful information on the spectral density of the ensemble in the large n limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large n limit. 1