Tracy-Widom GUE law and symplectic invariants

2010/11/05 by Borot, Gaetan, Eynard, Bertrand
#30Exx #33E17 #34E05 #60F10 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1011.1418

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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