In this talk, we focus on the interplay between the theory of integrable systems, and random matrix theory.
This connection was first realized by H. Spohn, who was able to compute the density of states for the Toda lattice by connecting it to the corresponding one of the Gaussian β ensemble, a well known random matrix model. The computation of this quantity enabled him to apply the theory of generalized hydrodynamics, so to compute the correlation functions for the Toda lattice.
In this talk, I consider another integrable model, namely the Ablowitz-Ladik lattice; I introduce the Generalized Gibbs ensemble for this lattice, and I relate it with the so-called Circular β ensemble, a classical random matrix model for unitary matrices. This allows us to compute explicitly the density of states for the Ablowitz-Ladik lattice in terms of the one of this random matrix ensemble.
This talk is mainly based on these two papers:

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

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