Discrete Integrable Systems and Random Lax Matrices

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 $$\beta $$ 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.

Large deviations for Ablowitz-Ladik lattice, and the Schur flow

“…In this subsection, we focus on the Volterra lattice and its relation with the Antisymmetric Gaussian β ensemble [20] in the high-temperature regime [26]. These two systems were considered in [31]. In this paper, the authors considered the classical Gibbs ensemble for the Volterra lattice and were able to compute the density of states for this model connecting it to the Antisymmetric Gaussian α ensemble [26], which is related to the classical β one introduced by Dumitriu and Forrester [20].…”

“…The Volterra lattice is a completely integrable system, and it admits several equivalents Lax representations, see e.g. [31,42,54]. We use the one presented in [31].…”

“…Specifically, the Toda lattice [70] and the Exponential Toda lattice [31] have as Lax matrix a periodic Jacobi matrix, the Volterra lattice [31] has an antisymmetric periodic one, the Ablowitz-Ladik lattice [2] and the Schur flow [29] have a periodic CMV one, and the family of Itoh-Narita-Bogoyavleskii [9] lattices have a bidiagonal periodic one.…”

“…For all these models, a large deviation principle for their mean density of states were developed in [35,53]. Furthermore, in [31] the authors were able to connect the classical Gibbs ensemble for the Exponential Toda lattice and the Volterra one to the Laguerre [19] and to the Antisymmetric β ensemble [20] respectively.…”

“…Remark 3.8. The definition of our Gibbs ensemble is slightly different from the one given in [31], indeed there the authors were considering just the classical Gibbs ensemble for this model, so the case P pxq " x{2.…”