Tracy-Widom GUE law and symplectic invariants

Details Tracy-Widom GUE law and symplectic invariants Tracy-Widom GUE law and symplectic invariants

2010-11-05

arxiv.org/abs/1011.1418v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

pdfLatex, 36 pages, 1 figure. v2: typos corrected, re-sectioning, a reference added

Scientific paper

We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curve \Sigma_{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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