Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GOE

TL;DR

This paper derives an Edgeworth expansion for the distribution of the largest eigenvalue in GOE matrices, providing correction terms to the Tracy-Widom distribution expressed via Painleve II functions.

Contribution

It introduces an Edgeworth type theorem for GOE eigenvalue distribution, extending the Tracy-Widom results with explicit correction terms.

Findings

Derived correction terms expressed through Painleve II functions

Extended Tracy-Widom distribution with Edgeworth expansion

Brief discussion on GSE case included

Abstract

In this paper we focus on the large n probability distribution function of the largest eigenvalue in the Gaussian Orthogonal Ensemble of n by n matrices (GOEn). We prove an Edgeworth type Theorem for the largest eigenvalue probability distribution function of GOEn. The correction terms to the limiting probability distribution are expressed in terms of the same Painleve II functions appearing in the Tracy-Widom distribution. We conclude with a brief discussion of the GSEn case.