In this paper, we prove a polynomial Central Limit Theorem for several integrable models, and for the β ensembles at high-temperature with polynomial potential. Furthermore, we are able to relate the mean values, the variances and the correlations of the moments of these integrable systems with one of the β ensembles. Moreover, we show that for several integrable models, the local functions' space-correlations decay exponentially fast.

We study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number N of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian beta-ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz-Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states.

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

Exploiting the theory of loop equations, we produced a unifying mechanism to characterize the mean density of states for the beta ensemble, with beta proportional to 1/N. We also characterize the moments and the covariances of monomial linear statistics through recurrence relations. Finally, we define the Gaussian antisymmetric alpha ensemble, and we computed its mean density of states and its mean spectral measure. From the explicit formula, we are able to supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain. 

We study the periodic Fermi-Pasta-Ulam-Tsingou system as a perturbation of the periodic Toda lattice equations. Exploiting this idea, we show that the Toda integral of motions are adiabatic invariants for the Fermi-Pasta-Ulam-Tsingou system in the thermodynamic limit. This result holds in probability with respect to the Gibbs measure of the system.

In recent years, a lot of effort has been put in describing the hydrodynamic behavior of integrable systems. In this paper, we describe such picture for the Volterra lattice. Specifically, we are able to explicitly compute the susceptibility matrix and the current-field correlation matrix in terms of the density of states of the Volterra lattice endowed with a Generalized Gibbs ensemble. Furthermore, we apply the theory of linear Generalized Hydrodynamics to describe the Euler scale behavior of the correlation functions. We anticipate that the solution to the Generalized Hydrodynamics equations develops shocks at $\xi_0 = x/t$; so this linear approximation does not fully describe the behavior of correlation functions. Intrigued but this fact, we performed several numerical investigations which show that, exactly when the solution to the hydrodynamic equations develops shock, the correlation functions show an highly oscillatory behavior. In view of this empirical observation, we believe that at this point $\xi_0$ the diffusive contribution are not sub-leading corrections to the ballistic transport, but they are of the same order.