Discrete integrable systems and random Lax matrices

TL;DR

This paper analyzes the spectral properties of Hamiltonian integrable systems with random initial data, revealing explicit density of states in certain models and numerical results for others, advancing understanding of their large-system behavior.

Contribution

It provides an exact description of the spectral density for specific integrable lattices and extends numerical analysis to generalized models, connecting integrable systems with random matrix theory.

Findings

Exact density of states for exponential Toda and Volterra lattices.

Numerical density of states for generalized lattices.

Explicit ground state density of states.

Abstract

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.