Saturation of number variance in embedded random matrix ensembles

TL;DR

This paper investigates the fluctuation properties of embedded random matrix ensembles of non-interacting particles, revealing saturation of number variance at large correlation lengths, a phenomenon previously known in integrable systems but now demonstrated in random matrix theory.

Contribution

It is the first to demonstrate saturation of number variance in embedded random matrix ensembles, bridging concepts from integrable systems and random matrix theory.

Findings

Number distributions follow Poisson statistics.

Number variance saturates at large correlation lengths.

Correlation functions are non-stationary in these ensembles.

Abstract

We study fluctuation properties of embedded random matrix ensembles of non-interacting particles. For ensemble of two non-interacting particle systems, we find that unlike the spectra of classical random matrices, correlation functions are non-stationary. In the locally stationary region of spectra, we study the number variance and the spacing distributions. The spacing distributions follow the Poisson statistics which is a key behavior of uncorrelated spectra. The number variance varies linearly as in the Poisson case for short correlation lengths but a kind of regularization occurs for large correlation lengths, and the number variance approaches saturation values. These results are known in the study of integrable systems but are being demonstrated for the first time in random matrix theory. We conjecture that the interacting particle cases, which exhibit the characteristics of classical random matrices for short correlation lengths, will also show saturation effects for large correlation lengths.