Random matrix ensembles associated with Lax matrices

TL;DR

This paper introduces a novel method to create random matrix ensembles linked to integrable systems, enabling analytical eigenvalue distribution calculations and revealing unique spectral statistics that often exhibit intermediate behavior.

Contribution

It proposes a new class of random matrix ensembles derived from Lax matrices of integrable systems, allowing exact spectral analysis and uncovering unusual intermediate spectral statistics.

Findings

Eigenvalue distributions can be computed analytically.

Spectral statistics often display intermediate behavior.

New examples of spectral statistics are rigorously demonstrated.

Abstract

A method to generate new classes of random matrix ensembles is proposed. Random matrices from these ensembles are Lax matrices of classically integrable systems with a certain distribution of momenta and coordinates. The existence of an integrable structure permits to calculate the joint distribution of eigenvalues for these matrices analytically. Spectral statistics of these ensembles are quite unusual and in many cases give rigorously new examples of intermediate statistics.