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

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