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

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

2010-12-13

arxiv.org/abs/1012.2752v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

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

Scientific paper

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.

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