Universality and critical behavior in the chiral two-matrix model

TL;DR

This paper investigates the universality and critical phenomena in the chiral two-matrix model, deriving asymptotics and limiting kernels using Riemann-Hilbert analysis, and extending known results to the chiral setting.

Contribution

It establishes the determinantal structure of singular values in the chiral two-matrix model and derives asymptotic behavior and critical kernels, extending non-chiral results to the chiral case.

Findings

Correlation kernel determined by a matrix-valued Riemann-Hilbert problem

Universality proven for quadratic potential W(y)=y^2/2+α y

Multi-critical limit described by Painlevé II related Riemann-Hilbert problem

Abstract

We study the chiral two-matrix model with polynomial potential functions $V$ and $W$, which was introduced by Akemann, Damgaard, Osborn and Splittorff. We show that the squared singular values of each of the individual matrices in this model form a determinantal point process with correlation kernel determined by a matrix-valued Riemann-Hilbert problem. The size of the Riemann-Hilbert matrix depends on the degree of the potential function $W$ (or $V$ respectively). In this way we obtain the chiral analogue of a result of Kuijlaars-McLaughlin for the non-chiral two-matrix model. The Gaussian case corresponds to $V,W$ being linear. For the case where $W(y)=y^2/2+\alpha y$ is quadratic, we derive the large $n$-asymptotics of the Riemann-Hilbert problem by means of the Deift-Zhou steepest descent method. This proves universality in this case. An important ingredient in the analysis is a third-order differential equation. Finally we show that if also $V(x)=x$ is linear, then a multi-critical limit of the kernel exists which is described by a $4\times 4$ matrix-valued Riemann-Hilbert problem associated to the Painlev\'e II equation $q"(x) = xq(x)+2q^3(x)-\nu-1/2$. In this way we obtain the chiral analogue of a recent result by Duits and the second author.