In the classical β-ensembles of random matrix theory, setting β = 2α/N and taking the N → ∞ limit gives a statistical state depending on α. Using the loop equations for the classical β-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading 1/N correction to the density. From earlier literature, the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying, in particular, the Gauss hypergeometric differential equation determining the Stieltjes transform of the limiting density in the Jacobi case. Our characterization of the moments and covariances of monomial linear statistics is through recurrence relations. We also extend recent work, which begins with the β-ensembles in the high-temperature limit and constructs a family of tridiagonal matrices referred to as α-ensembles, obtaining a random anti-symmetric tridiagonal matrix with i.i.d. (Independent Identically Distributed) gamma distributed random variables. From this, we can supplement analytic results obtained by Dyson in the study of the so-called type I disordered chain.

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by $$N \gg 1$$ N ≫ 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $$\beta ^{-1}$$ β - 1 . Given a fixed $${1\le m \ll N}$$ 1 ≤ m ≪ N , we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $$\beta $$ β , for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

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 $$\beta $$ β -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.

In this paper, we study tridiagonal random matrix models related to the classical β-ensembles (Gaussian, Laguerre, and Jacobi) in the high-temperature regime, i.e., when the size N of the matrix tends to infinity with the constraint that βN = 2 α constant, α > 0. We call these ensembles the Gaussian, Laguerre, and Jacobi α-ensembles, and we prove the convergence of their empirical spectral distributions to their mean densities of states, and we compute them explicitly. As an application, we explicitly compute the mean density of states of the Lax matrix of the Toda lattice with periodic boundary conditions with respect to the Gibbs ensemble.

In this manuscript we study tridiagonal random matrix models related to the classical β-ensembles (Gaussian, Laguerre, Jacobi) in the high temperature regime, i.e. when the size N of the matrix tends to infinity with the constraint that βN " 2α constant, α ą 0. We call these ensembles the Gaussian, Laguerre and Jacobi α-ensembles and we prove the convergence of their empirical spectral distributions to their mean densities of states and we compute them explicitly. As an application we explicitly compute the mean density of states of the Lax matrix of the Toda lattice with periodic boundary conditions with respect to the Gibbs ensemble.