Large Deviations for Ablowitz-Ladik lattice, and the Schur flow

TL;DR

This paper establishes large deviations principles for the empirical measures of the Ablowitz-Ladik lattice and Schur flow, linking their limits to classical ensembles and proving almost sure convergence.

Contribution

It derives large deviations principles for these integrable systems and characterizes their limits via classical random matrix ensembles.

Findings

Large deviations principles are established for the ensembles.

Almost sure convergence of empirical measures is proved.

Limits are characterized by the Circular and Jacobi beta ensembles.

Abstract

We consider the Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, and the Schur flow. We derive large deviations principles for the distribution of the empirical measures of the equilibrium measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limit in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble respectively.