Gap probabilities in the finite and scaled Cauchy random matrix ensembles

TL;DR

This paper derives exact formulas for gap probabilities in finite and scaled Cauchy random matrix ensembles, linking them to Painlevé transcendents and providing new insights into their spectral properties.

Contribution

It provides explicit expressions for gap probabilities in finite and scaled ensembles, connecting them to Painlevé equations and revealing their scaling behavior.

Findings

Exact formulas for gap probabilities in finite ensembles.

Connection of scaled gap probabilities to Painlevé-V transcendents.

Simplified expression for the spacing probability density function.

Abstract

The probabilities for gaps in the eigenvalue spectrum of finite $ N\times N $ random unitary ensembles on the unit circle with a singular weight, and the related hermitian ensembles on the line with Cauchy weight, are found exactly. The finite cases for exclusion from single and double intervals are given in terms of second order second degree ODEs which are related to certain \mbox{Painlev\'e-VI} transcendents. The scaled cases in the thermodynamic limit are again second degree and second order, this time related to \mbox{Painlev\'e-V} transcendents. Using transformations relating the second degree ODE and transcendent we prove an identity for the scaled bulk limit which leads to a simple expression for the spacing p.d.f. We also relate all the variables appearing in the Fredholm determinant formalism to particular \mbox{Painlev\'e} transcendents, in a simple and transparent way, and exhibit their scaling behaviour.