Nonlinear PDEs for Fredholm determinants arising from string equations

Abstract

String equations related to 2D gravity seem to provide, quite naturally and systematically, integrable kernels, in the sense of Its-Izergin-Korepin and Slavnov. Some of these kernels (besides the "classical" examples of Airy and Pearcey) have already appeared in random matrix theory and they have a natural Wronskian structure, given by one of the operators in the string relation $[L^{\pm },Q^{\pm }]=\pm 1$, namely $L^{\pm }$. The kernels are intimately related to wave functions for Gel'fand-Dickey reductions of the KP hierarchy. The Fredholm determinants of these kernels also satisfy Virasoro constraints leading to PDEs for their log derivatives, and these PDEs depend explicitly on the solutions of Painlevé-like systems of ODEs equivalent to the relevant string relations. We give some examples coming from critical phenomena in random matrix theory (higher order Tracy-Widom distributions) and statistical mechanics (Ising models).

Accepted for publication on the AMS Contemporary Mathematics Series, 36 pages

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• Dubrovin's superpotential as a global spectral curve
• Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type
• Non-commutative lattice modified Gel'fand-Dikii systems
• Discrete KP equation with self-consistent sources
• Quantum curves and conformal field theory
• Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion
• Log topological recursion through the prism of $x-y$ swap
• Integrable structure of products of finite complex Ginibre random matrices
• Classification of algebraic solutions of irregular Garnier systems