Finite n Largest Eigenvalue Probability Distribution Function of Gaussian Ensembles

TL;DR

This paper derives the finite n probability distribution functions for the largest eigenvalue in Gaussian Orthogonal and Symplectic Ensembles, including correction terms expressed via Painleve II functions, extending Tracy-Widom results.

Contribution

It provides explicit finite n formulas for the largest eigenvalue distribution in Gaussian ensembles and proves an Edgeworth expansion for the symplectic case.

Findings

Finite n distribution functions derived for GOE and GSE

Correction terms expressed with Painleve II functions

Edgeworth expansion proved for GSE largest eigenvalue

Abstract

In this paper we focus on the finite n probability distribution function of the largest eigenvalue in the classical Gaussian Ensemble of n by n matrices (GEn). We derive the finite n largest eigenvalue probability distribution function for the Gaussian Orthogonal and Symplectic Ensembles and also prove an Edgeworth type Theorem for the largest eigenvalue probability distribution function of Gaussian Symplectic Ensemble. The correction terms to the limiting probability distribution are expressed in terms of the same Painleve II functions appearing in the Tracy-Widom distribution.