Weak and strong confinement in the Freud random matrix ensemble and gap probabilities

TL;DR

This paper studies eigenvalue behavior near zero in the Freud random matrix ensemble, revealing a transition from sine process to a beta-dependent process as the confinement parameter varies, and computes gap probabilities and asymptotics.

Contribution

It characterizes the local eigenvalue statistics across a phase transition in the Freud ensemble and derives explicit asymptotics for gap probabilities and integrals in different regimes.

Findings

Eigenvalue behavior transitions at beta=1 from sine process to beta-dependent process.

First two terms of large gap probability in the weak confinement regime (0<beta<1).

Asymptotic constant for Freud multiple integral for beta≥1.

Abstract

The Freud ensemble of random matrices is the unitary invariant ensemble corresponding to the weight $\exp(-n |x|^{\beta})$, $\beta>0$, on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in $\beta$. If $\beta\ge 1$, it is described by the standard sine process. Below the critical value $\beta=1$, it is described by a process depending on the value of $\beta$, and we determine the first two terms of the large gap probability in it. This so called weak confinement range $0<\beta<1$ corresponds to the Freud weight with the indeterminate moment problem. We also find the multiplicative constant in the asymptotic expansion of the Freud multiple integral for $\beta\ge 1$.