Integrable Lattices: Random Matrices and Random Permutations

TL;DR

This paper surveys recent advances linking random matrices and permutations to integrable systems, showing how matrix integrals serve as tau-functions for various lattices and PDEs, and satisfy Virasoro constraints.

Contribution

It demonstrates the connection between matrix integrals, tau-functions, and integrable systems, highlighting new integrable structures in probabilistic models.

Findings

Matrix integrals are natural tau-functions for integrable lattices.

These integrals satisfy Virasoro constraints leading to differential equations.

Probabilistic problems relate to integrable PDEs like KdV.

Abstract

These lectures present a survey of recent developments in the area of random matrices (finite and infinite) and random permutations. These probabilistic problems suggest matrix integrals (or Fredholm determinants), which arise very naturally as integrals over the tangent space to symmetric spaces, as integrals over groups and finally as integrals over symmetric spaces. An important part of these lectures is devoted to showing that these matrix integrals, upon apropriately adding time-parameters, are natural tau-functions for integrable lattices, like the Toda, Pfaff and Toeplitz lattices, but also for integrable PDE's, like the KdV equation. These matrix integrals or Fredholm determinants also satisfy Virasoro constraints, which combined with the integrable equations lead to (partial) differential equations for the original probabilities.