Some distribution functions of the Laguerre ensemble

TL;DR

This paper derives closed-form integrals for the smallest eigenvalue distribution in Laguerre ensembles, computes two-smallest-eigenvalue distributions using Dyson's approximation, and explores high-order effects in various scaling limits.

Contribution

It provides new closed-form integral representations and extends the analysis of eigenvalue distributions in Laguerre ensembles beyond previous results.

Findings

Closed-form integrals for smallest eigenvalue distribution

Two-smallest-eigenvalue distribution computed via Dyson's approximation

High-order contributions reveal entropic effects in scaling limits

Abstract

In this paper we revisit the smallest-eigenvalue distribution of the Laguerre ensembles by presenting in closed form certain integrals obtained previously. With this information we compute, using Dyson's continuum approximation, the two-smallest-eigenvalue distribution of the Laguerre ensemble. High-order contributions to the free energies describing these two probabilities in certain scaling limits or \beta\neq 2, which can be interpreted in this context as an entropic effect, are found.