Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis

TL;DR

This paper analyzes the eigenvalue statistics of a two matrix model using Riemann-Hilbert steepest descent methods, revealing universal behavior in the large matrix limit for specific polynomial potentials.

Contribution

It provides a steepest descent analysis of a matrix-valued Riemann-Hilbert problem for biorthogonal polynomials in a two matrix model with quartic potential, establishing universality results.

Findings

Limiting behavior of the correlation kernel in the large n limit

Global and local eigenvalue statistics characterized in the one-cut case

Introduction of a vector equilibrium problem with external field and constraint

Abstract

The eigenvalue statistics of a pair $(M_1,M_2)$ of $n\times n$ Hermitian matrices taken random with respect to the measure $$\frac{1}{Z_n}\exp\big(-n\Tr (V(M_1)+W(M_2)-\tau M_1M_2)\big) {\rm d}M_1 {\rm d} M_2 $$ can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest descent analysis of a $4 \times 4$ matrix-valued Riemann-Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case $W(y)=y^4/4$ and $V$ an even polynomial. As a result we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of $M_1$ (when averaged over $M_2$) in the global and local regime as $n\to \infty$ in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.