Guido Mazzuca: Large deviation principle for the Ablowitz-Ladik lattice and the circular beta ensemble at high temperature

Abstract.

This presentation delves into the fascinating relationship between integrable systems theory and random matrix theory. In particular, I show a method to describe the eigenvalue density of the Ablowitz-Ladik lattice with random initial data sampled from a Generalized Gibbs ensemble. This characterization is achieved by employing a large deviation principle (LDP). Additionally, we establish a connection between this LDP and the one of the Circular beta ensemble in the high temperature regime. As a result, we can explicitly compute the eigenvalue density of the Ablowitz-Ladik lattice using the density of the random matrix ensemble. This talk is mainly based on

G. Mazzuca, and R. Memin: Large Deviations for Ablowitz-Ladik lattice, and the Schur flow. Electronic Journal of Probability. DOI: 10.1214/23-EJP941. ‚

G. Mazzuca, and T. Grava: Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, circular beta- ensemble and double confluent Heun equation. Communication in Mathematical Physics. DOI: 10.1007/s00220-023-04642-8.