Random Matrices with Merging Singularities and the Painlev\'e V Equation

TL;DR

This paper investigates the asymptotic behavior of certain random matrix ensembles with merging singularities, revealing a phase transition described by Painlevé V equations and identifying algebraic solutions for specific parameters.

Contribution

It introduces a double scaling limit analysis of random matrices with merging singularities, connecting the transition to Painlevé V equations and explicit algebraic solutions.

Findings

Identification of a phase transition described by Painlevé V

Explicit asymptotics for partition function and correlation kernel

Algebraic solutions for special parameter values

Abstract

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $\alpha>-1/2$ and $t\in\mathbb R$, in double scaling limits where $n\to\infty$ and simultaneously $t\to 0$. If $t$ is proportional to $1/n^2$, a transition takes place which can be described in terms of a family of solutions to the Painlev\'e V equation. These Painlev\'e solutions are in general transcendental functions, but for certain values of $\alpha$, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.