The distribution of the ratio of consecutive level spacings in random matrix ensembles

TL;DR

This paper derives and validates analytical expressions for the distribution of ratios of consecutive level spacings in random matrix ensembles, useful for analyzing spectral properties in many-body physics and number theory.

Contribution

It provides new analytical formulas for the ratio distribution in classical random matrix ensembles, improving accuracy with polynomial expansions and demonstrating applications to quantum models and zeta zeros.

Findings

Derived accurate ratio distribution formulas for random matrix ensembles.

Validated formulas against numerical and exact results, showing high accuracy.

Applied the results to quantum many-body systems and Riemann zeta zeros.

Abstract

We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.