Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges

TL;DR

This paper investigates the universal behavior of Cauchy transforms of orthogonal polynomials related to ratios of characteristic polynomials in large random Hermitian matrices, across different spectral regimes.

Contribution

It extends universality results to Cauchy transforms at spectral edges and demonstrates they satisfy the same differential equations as orthogonal polynomials.

Findings

Universality holds for Cauchy transforms at hard, bulk, and soft spectral edges.

Cauchy transforms obey the same differential equations as orthogonal polynomials.

Results apply to a broad class of weight functions.

Abstract

It has been shown recently [10] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random NxN Hermitian matrices. Our main goal is to investigate the issue of universality of large N asymptotics for those Cauchy transforms for a wide class of weight functions. Our analysis covers three different scaling regimes: the "hard edge", the "bulk" and the "soft edge" of the spectrum, thus extending the earlier results known for the bulk. The principal tool is to show that for finite matrix size N the auxiliary "wave functions" associated with the Cauchy transforms obey the same second order differential equation as those associated with the orthogonal polynomials themselves.