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arXiv · 1012.2752

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

Gaetan Borot·Bertrand Eynard
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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.

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Gaetan Borot, Bertrand Eynard. 2010-12-13. The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants. https://arxiv.org/abs/1012.2752

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