Painlev\'e transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles

TL;DR

This paper evaluates the scaled distribution of the smallest eigenvalue in Laguerre orthogonal and symplectic ensembles using Painlevé V transcendent, extending previous work on the Laguerre unitary ensemble.

Contribution

It introduces a Painlevé V transcendent approach to characterize the smallest eigenvalue distribution in orthogonal and symplectic ensembles, building on Tracy and Widom's framework.

Findings

Derived explicit Painlevé V representations

Extended Tracy-Widom results to orthogonal and symplectic cases

Provided formulas for scaled eigenvalue distributions

Abstract

The scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles is evaluated in terms of a Painlev\'e V transcendent. This same Painlev\'e V transcendent is known from the work of Tracy and Widom, where it has been shown to specify the scaled distribution of the smallest eigenvalue in the Laguerre unitary ensemble. The starting point for our calculation is the scaled $k$-point distribution of every odd labelled eigenvalue in two superimposed Laguerre orthogonal ensembles.