The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants

TL;DR

This paper links the asymptotic expansion of the Tracy-Widom GUE law to symplectic invariants derived from a specific plane curve related to random matrix eigenvalue distributions near a hard edge.

Contribution

It establishes a novel connection between Painlevé II related integrable systems and symplectic invariants for describing eigenvalue distribution asymptotics.

Findings

Asymptotic expansion of Tracy-Widom GUE law expressed via symplectic invariants.

Relation between integrable systems and geometric invariants in random matrix theory.

Provides a new geometric perspective on eigenvalue distribution tails.

Abstract

We establish the relation between two objects: an integrable system related to Painlev\'e II equation, and the symplectic invariants of a certain plane curve S(TW). This curve describes the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This shows that the s -> -infinity asymptotic expansion of Tracy-Widow law F_{GUE}(s), governing the distribution of the maximal eigenvalue in hermitian random matrices, is given by symplectic invariants.