Multi-critical unitary random matrix ensembles and the general Painleve II equation

TL;DR

This paper investigates a class of unitary random matrix ensembles with a focus on the local eigenvalue behavior near the origin, revealing connections to a special solution of the Painleve II equation and establishing the absence of real poles for certain parameters.

Contribution

The study constructs a local parametrix for the Riemann-Hilbert problem using Painleve II solutions and proves the pole-free nature of these solutions for specific parameter ranges.

Findings

The local eigenvalue correlation kernel converges to a limit described by Painleve II.

The asymptotics of orthogonal polynomial recurrence coefficients are expressed via Painleve II solutions.

The special Painleve II solutions have no real poles for a> -1/2.

Abstract

We study unitary random matrix ensembles of the form $Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \Tr V(M)}dM$, where $\alpha>-1/2$ and $V$ is such that the limiting mean eigenvalue density for $n,N\to\infty$ and $n/N\to 1$ vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight $|x|^{2\alpha}e^{-NV(x)}$. Here the main focus is on the construction of a local parametrix near the origin with $\psi$-functions associated with a special solution $q_\alpha$ of the Painlev\'e II equation $q''=sq+2q^3-\alpha$. We show that $q_\alpha$ has no real poles for $\alpha > -1/2$, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of $q_\alpha$ in the double scaling limit.