Classical integrability for beta-ensembles and general Fokker-Planck equations

TL;DR

This paper proves classical integrability of beta-ensembles for even integer beta by constructing explicit Lax pairs, linking solutions of Fokker-Planck equations to eigenvectors, and solving related nonlinear PDEs.

Contribution

It demonstrates classical integrability for even integer beta-ensembles by explicitly constructing Lax pairs and connecting Fokker-Planck solutions to integrable systems.

Findings

Lax pairs are explicitly constructed for even integer beta.

Solutions of Fokker-Planck equations relate to eigenvectors of Lax pairs.

Solution for beta related to quantum Painlevé II involves Calogero systems.

Abstract

Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlev\'e Hamiltonians. Here we demonstrate that, at least for {\it even integer} beta, these systems are classically integrable, e.g. there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. The explicit finding of the Lax pair depends on finding a solution of a governing system -- a closed system of two nonlinear PDEs of hydrodynamic type. This result suggests that there must be a solution for all values of beta. We find the solution of this system for even integer beta in the particular case of quantum Painlev\'e II related to the soft edge of the spectrum for beta-ensembles. The solution is given in terms of Calogero system of $\beta/2$ particles in an additional time-dependent potential. Thus, we find another situation where quantum integrability is reduced to classical integrability.