Orthogonal Polynomials and Exact Correlation Functions for Two Cut Random Matrix Models

TL;DR

This paper derives exact eigenvalue correlation functions for large N hermitian two-cut random matrix models, using orthogonal polynomials to analyze universal behaviors in eigenvalue distributions.

Contribution

It introduces an asymptotic form for orthogonal polynomials in symmetric two-cut models, enabling explicit kernel expressions and analysis of universality in eigenvalue correlations.

Findings

Explicit kernel expression for large N two-cut models

Demonstration of universality in local and global correlators

Extraction of oscillating and smooth parts of the two-point correlator

Abstract

Exact eigenvalue correlation functions are computed for large $N$ hermitian one-matrix models with eigenvalues distributed in two symmetric cuts. An asymptotic form for orthogonal polynomials for arbitrary polynomial potentials that support a $Z_2$ symmetric distribution is obtained. This results in an exact explicit expression for the kernel at large $N$ which determines all eigenvalue correlators. The oscillating and smooth parts of the two point correlator are extracted and the universality of local fine grained and smoothed global correlators is established.