Large deviations for generalized Gibbs ensembles of the classical Toda chain

Abstract: We derive large deviations principles for the distribution of the empirical measure of the equilibrium measure for the Generalized Gibbs ensembles of the classical Toda chain introduced in [11]. We deduce its almost sure convergence and characterize its limit in terms of the limiting measure of Beta-ensembles. Our results apply to general smooth potentials.