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arXiv · 1011.1418

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 Painleve II equation, and the symplectic invariants of a certain plane curve Σ_{TW} describing the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This explains directly how the Tracy-Widow law F_{GUE}, governing the distribution of the maximal eigenvalue in hermitian random matrices, can also be recovered from symplectic invariants.

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Gaetan Borot, Bertrand Eynard. 2010-11-22. Tracy-Widom GUE law and symplectic invariants. https://arxiv.org/abs/1011.1418

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