Right tail expansion of Tracy-Widom beta laws

TL;DR

This paper derives the large deviation function for the maximum eigenvalue in Gaussian beta ensembles and provides an all-order asymptotic expansion of the right tail Tracy-Widom beta laws for all positive beta, using loop equations and double scaling limits.

Contribution

It introduces a method to compute the right tail large deviations and asymptotic expansions for Tracy-Widom beta laws for all positive beta, extending previous results.

Findings

Derived the large deviation function for the maximum eigenvalue.

Obtained the all-order asymptotic expansion of Tracy-Widom beta laws.

Unified the derivation for all positive beta values.

Abstract

Using loop equations, we compute the large deviation function of the maximum eigenvalue to the right of the spectrum in the Gaussian beta matrix ensembles, to all orders in 1/N. We then give a physical derivation of the all order asymptotic expansion of the right tail Tracy-Widom beta laws, for all positive beta, by studying the double scaling limit.