Lax random matrices from Calogero systems

TL;DR

This paper investigates the eigenvalue density of random matrices derived from the Lax matrices of Calogero integrable systems, linking integrable models with generalized random matrix ensembles and confirming findings through simulations.

Contribution

It introduces a novel class of random matrices from integrable systems and analyzes their eigenvalue distributions, expanding understanding of integrable models and their hydrodynamic descriptions.

Findings

Eigenvalue density characterized for matrices from Calogero models

Mapping to generalized random matrix ensembles established

Monte-Carlo simulations confirm theoretical predictions

Abstract

We study a class of random matrices arising from the Lax matrix structure of classical integrable systems, particularly the Calogero family of models. Our focus is the density of eigenvalues for these random matrices. The problem can be mapped to analyzing the density of eigenvalues for generalized versions of conventional random matrix ensembles, including a modified form of the log-gas. The mapping comes from the underlying integrable structure of these models. Such deep connection is confirmed by extensive Monte-Carlo simulations. Thereby we move forward not only in terms of understanding such class of random matrices arising from integrable many-body systems, but also by providing a building block for the generalized hydrodynamic description of integrable systems.