Large scale correlations in normal and general non-Hermitian matrix ensembles

TL;DR

This paper analyzes large-scale eigenvalue correlations in normal and non-Hermitian matrix ensembles, revealing universal behaviors and providing explicit formulas for multi-point correlations based on boundary value problems.

Contribution

It introduces a method to compute multi-point correlation functions for complex eigenvalues in large matrix ensembles, emphasizing universality and boundary problem solutions.

Findings

Correlation functions depend only on the eigenvalue support boundary.

Explicit formulas for two-, three-, and four-point correlations are derived.

Correlation functions are expressed via Dirichlet Green functions of the domain complement.

Abstract

We compute the large scale (macroscopic) correlations in ensembles of normal random matrices with an arbitrary measure and in ensembles of general non-Hermition matrices with a class of non-Gaussian measures. In both cases the eigenvalues are complex and in the large $N$ limit they occupy a domain in the complex plane. For the case when the support of eigenvalues is a connected compact domain, we compute two-, three- and four-point connected correlation functions in the first non-vanishing order in 1/N in a manner that the algorithm of computing higher correlations becomes clear. The correlation functions are expressed through the solution of the Dirichlet boundary problem in the domain complementary to the support of eigenvalues. The two-point correlation functions are shown to be universal in the sense that they depend only on the support of eigenvalues and are expressed through the Dirichlet Green function of its complement.