Hyperbolic disordered ensembles of random matrices

TL;DR

This paper introduces a method to generate random matrices with spectral properties transitioning from semi-circle to Gaussian-like distributions, revealing a shift from Wigner-Dyson to Poisson statistics and exhibiting non-ergodic behavior.

Contribution

It presents a novel ensemble of random matrices based on dividing Gaussian matrices by a positive random variable, demonstrating a spectral transition and non-ergodic fluctuations.

Findings

Spectral density transitions from semi-circle to Gaussian-like.

Local fluctuations shift from Wigner-Dyson to Poisson statistics.

Number variance shows large fluctuations indicating non-ergodicity.

Abstract

Using the simple procedure, recently introduced, of dividing Gaussian matrices by a positive random variable, a family of random matrices is generated characterized by a behavior ruled by the generalized hyperbolic distribution. The spectral density evolves from the semi-circle law to a Gaussian-like behavior while concomitantly the local fluctuations show a transition from the Wigner-Dyson to the Poisson statistics. Long range statistics such as number variance exhibit large fluctuations typical of non-ergodic ensembles.