Conformal Deformation from Normal to Hermitian Random Matrix Ensembles

TL;DR

This paper explores how normal random matrix ensembles can be deformed conformally into Hermitian ensembles, analyzing eigenvalue statistics and conditions under which they resemble Wigner ensembles as matrix size grows large.

Contribution

It introduces a framework for conformal deformation from normal to Hermitian ensembles and provides conditions for these to become Wigner ensembles in the large N limit.

Findings

Eigenvalue distributions are uniform within regions bounded by polynomial curves.

Sufficient conditions are established for Hermitian ensembles to approximate Wigner ensembles.

The study connects conformal deformations with classical random matrix models.

Abstract

We investigate the eigenvalues statistics of ensembles of normal random matrices when their order N tends to infinite. In the model the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve. We study the conformal deformations of normal random ensembles to Hermitian random ensembles and give sufficient conditions for the latter to be a Wigner ensemble.