A Simple Approach to Global Regime of the Random Matrix Theory

TL;DR

This paper introduces a straightforward method for asymptotic analysis of eigenvalue distributions in large random matrices, utilizing resolvent identities and perturbation theory to derive functional equations and convergence bounds.

Contribution

It presents a novel, simple approach to compute moments and analyze the limiting eigenvalue distribution of large random matrices.

Findings

Derived functional equations for Stieltjes transforms.

Established bounds for convergence rates.

Unified approach applicable to various random matrix ensembles.

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.