Random matrices applications to soft spectra

TL;DR

This paper explores the application of random matrix theory to analyze the spectral properties of systems with weak or strong disorder, revealing a fluctuation scaling mechanism in long-range statistics.

Contribution

It extends standard RMT methods to include long spectra analysis, uncovering a fluctuation scaling mechanism in spectra with disorder.

Findings

Spectra can show quenched local statistics with fluctuating long-range statistics.

Long-range statistics follow the Taylor law, indicating a fluctuation scaling mechanism.

Disorder type influences the spectral statistical behavior.

Abstract

It recently has been found that methods of the statistical theories of spectra can be a useful tool in the analysis of spectra far from levels of Hamiltonian systems. Several examples originate from areas, such as quantitative linguistics and polymers. The purpose of the present study is to deepen this kind of approach by performing a more comprehensive spectral analysis that measures both the local and long-range statistics. We have found that, as a common feature, spectra of this kind can exhibit a situation in which local statistics are relatively quenched while the long range ones show large fluctuations. By combining extensions of the standard Random Matrix Theory (RMT) and considering long spectra, we demonstrate that this phenomenon occurs when weak disorder is introduced in a RMT spectrum or when strong disorder acts in a Poisson regime. We show that the long-range statistics follow the Taylor law, which suggests the presence of a fluctuation scaling (FS) mechanism in this kind of spectra.