Level-Spacing Distributions and the Bessel Kernel

TL;DR

This paper investigates the distribution of eigenvalue spacings in certain matrix ensembles, expressing these distributions via Bessel functions and deriving related differential equations, including Painleve equations.

Contribution

It introduces a new representation of level spacing distributions using Bessel kernels and derives associated PDEs, connecting random matrix theory with integrable systems.

Findings

Distribution expressed as Fredholm determinant with Bessel kernel

Derived PDEs for the logarithmic derivative of the Fredholm determinant

Identified the determinant as a Painleve tau function in the single-interval case

Abstract

The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is expressible in terms of Bessel functions of order $\alpha$. We derive a system of partial differential equations associated with the logarithmic derivative of this Fredholm determinant when the underlying domain is a union of intervals. In the case of a single interval this Fredholm determinant is a Painleve tau function.