Integrable random matrix ensembles

TL;DR

This paper introduces new integrable random matrix ensembles that interpolate between Wigner-Dyson and Poisson statistics, with analytically computable eigenvalue distributions derived from classical integrable systems.

Contribution

It presents a novel construction of random matrix ensembles based on integrable N-body systems, enabling explicit calculation of spectral statistics.

Findings

Derived formulas for spacing distributions.

Calculated level compressibility.

Established intermediate spectral statistics.

Abstract

We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner-Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to integrability of the underlying system. Formulas for spacing distributions and level compressibility are obtained for various instances of such ensembles.