Level Spacing Distribution of Critical Random Matrix Ensembles

TL;DR

This paper analyzes the eigenvalue spacing distribution in certain critical random matrix ensembles, revealing a universal kernel that interpolates between Wigner-Dyson and Poisson statistics, expressed via Painleve VI functions.

Contribution

It introduces a universal asymptotic kernel for critical ensembles with slowly growing potentials and derives an explicit eigenvalue spacing distribution using Painleve VI transcendents.

Findings

Universal kernel interpolates between Wigner-Dyson and Poisson statistics.

Eigenvalue spacing distribution expressed via Painleve VI functions.

Asymptotic behavior applies to ensembles with small deformation parameters.

Abstract

We consider unitary invariant random matrix ensembles which obey spectral statistics different from the Wigner-Dyson, including unitary ensembles with slowly (~(log x)^2) growing potentials and the finite-temperature fermi gas model. If the deformation parameters in these matrix ensembles are small, the asymptotically translational-invariant region in the spectral bulk is universally governed by a one-parameter generalization of the sine kernel. We provide an analytic expression for the distribution of the eigenvalue spacings of this universal asymptotic kernel, which is a hybrid of the Wigner-Dyson and the Poisson distributions, by determining the Fredholm determinant of the universal kernel in terms of a Painleve VI transcendental function.