Joint probability densities of level spacing ratios in random matrices

TL;DR

This paper derives analytical joint probability densities for ratios of level spacings in finite-size random matrices, providing explicit formulas for specific ensembles and validating them against simulations and known data.

Contribution

It introduces new analytical formulas for joint probability densities of level spacing ratios in finite random matrices, including generalizations like overlapping ratios.

Findings

Analytical formulas match numerical simulations.

Explicit calculations for beta-Hermite and beta-Laguerre ensembles.

Results agree with quantum many-body and Riemann zeta zeros data.

Abstract

We calculate analytically, for finite-size matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential. We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the beta-Hermite and the beta-Laguerre cases, for which we offer explicit calculations for small N. The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum many-body lattice model and zeros of the Riemann zeta function.