Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited

TL;DR

This paper provides refined asymptotic expansions for the largest eigenvalue distribution of the Gaussian Unitary Ensemble, offering a new proof of the Edgeworth expansion and discussing extensions to other Gaussian ensembles.

Contribution

It derives detailed resolvent and kernel expansions at the spectrum edge and offers an alternative proof of the Edgeworth expansion for GUE largest eigenvalue distribution.

Findings

Derived resolvent and kernel expansions at the spectrum edge.

Provided an alternative proof of the Edgeworth expansion for GUE.

Discussed potential extensions to GOE and GSE.

Abstract

We derive expansions of the resolvent Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we give another proof of the derivation of an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn. We conclude with a brief discussion on the derivation of the probability distribution function of the corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and Gaussian Symplectic Ensembles (GSEn).