Universality in the two matrix model with a monomial quartic and a general even polynomial potential

TL;DR

This paper investigates the asymptotic eigenvalue behavior of a two matrix model with a quartic monomial and a general even polynomial potential, extending existing results to multi-interval support cases.

Contribution

It extends the analysis of eigenvalue correlations in the two matrix model to cases with multi-interval support using Riemann-Hilbert problem techniques.

Findings

Extended the correlation kernel analysis to multi-interval support cases.

Constructed a parametrix using theta functions for the Riemann-Hilbert problem.

Proved the well-definedness of the parametrix via theta divisor analysis.

Abstract

In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with a quartic monomial and a general even polynomial potential. We studied the correlation kernel for the eigenvalues of one of the matrices in asymptotic limit. We extended the results of Duits and Kuijlaars to the case when the limiting eigenvalue density for one of the matrices is supported on multiple intervals. The results are achieved by constructing the parametrix to a Riemann-Hilbert problem obtained by Duits and Kuijlaars with theta functions and then showing that this parametrix is well-defined by studying the theta divisor.