Tracy-Widom GUE law and symplectic invariants

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.

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

References in corpus (7)

• Level-Spacing Distributions and the Airy Kernel
• All genus correlation functions for the hermitian 1-matrix model
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Algebraic methods in random matrices and enumerative geometry
• Determinantal formulae and loop equations
• Universal scaling limits of matrix models, and (p,q) Liouville gravity

Cited by in corpus (2)

• Quantum Curve and the First Painlevé Equation
• The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants