Nonlinear Lattices and Random Matrix

In this thesis, we study problems related to statistical properties of integrable and non-integrable Hamiltonian system, focusing on their relations with random matrix theory. First, we consider the harmonic chain with short-range interactions. Exploiting the rich theory of circulant and Toeplitz matrices, we are able to explicitly compute the correlation functions for this system. Further, applying the so-called steepest descent method, we compute their long time asymptotic. In the main part of the thesis, we focus on the interplay between Random Matrix theory and integrable Hamiltonian system. Specifically, we introduce some new tridiagonal random matrix ensembles that we name $alpha$ ensembles, and we compute their mean density of states. These random matrix models are related to the classical beta ones in the high temperature regime. Moreover, they are also connected to the Toda and the Ablowitz-Ladik lattice, indeed applying our result on the $alpha$ ensembles, we are able to compute the mean density of states of the Lax matrices of these two lattices. Next, we focus on the Fermi-Pasta-Ulam-Tsingou (FPUT) system, a non-integrable lattice. We show that the integrals of motion of the Toda lattice are adiabatic invariants, namely statistically almost conserved quantities, for the FPUT system for a time-scale of order $eta^{1-2arepsilon}$, here $arepsilon>0$, and $eta$ is the inverse of the temperature. Moreover, we show that some special linear combinations of the normal modes are adiabatic invariants for the Toda lattice, for all times, and for the FPUT, for times of order $eta^{1-2arepsilon}$. Finally, we consider the classical beta ensembles in the high temperature regime. We compute their mean density of states, making use of the so-called loop equations. Exploiting this formalism, we are able to compute the moments and the linear statistic covariance through recurrence relations. Further, we identify a new $alpha$ ensemble, which is related to Dyson's study of a disordered chain. Our analysis supplement the results contained in Dyson's work.