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

Abstract

We establish the relation between two objects: an integrable system related to Painlevé 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.

Short version of 1011.1418, centered on Tracy-Widom law. 15 pages

References in corpus (4)

• Distribution functions for largest eigenvalues and their applications
• Algebraic methods in random matrices and enumerative geometry
• Determinantal formulae and loop equations
• Tracy-Widom GUE law and symplectic invariants