Random matrix ensembles from nonextensive entropy

TL;DR

This paper introduces a family of random matrix ensembles derived from nonextensive entropy S_q, which generalize Gaussian ensembles and exhibit power-law tails and correlations for q ≠ 1, revealing new statistical behaviors.

Contribution

It constructs and analyzes nonextensive random matrix ensembles based on S_q, extending classical Gaussian ensembles to include correlated elements and non-Gaussian distributions.

Findings

For q<1, ensembles have compact support and Gaussian-like fluctuations.

For q>1, ensembles display power-law tails and non-Wigner-Dyson spacing distributions.

Numerical results show long-tailed spacing distributions close to 2x2 case, generalizing Wigner's surmise.

Abstract

The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, S_{BGS} = - \int d{\bf H} [P({\bf H})] \ln [P({\bf H})], with suitable constraints. Here we construct and analyze random-matrix ensembles arising from the generalized entropy S_q = (1 - \int d{\bf H} [P({\bf H})]^q)/(q-1) (thus S_1=S_{BGS}). The resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form. Making q -> 1 recovers the Gaussian ensembles. If q \ne 1, the joint probability distributions P(\bf H) cannot be factorized, i.e., the matrix elements of \bf H are correlated. In the limit of large matrices two different regimes are observed. When q<1, P(\bf H) has compact support, and the fluctuations tend asymptotically to those of the Gaussian ensembles. Anomalies appear for q>1: Both P(\bf H) and the marginal distributions P(H_{ij}) show power-law tails. Numerical analyses reveal that the nearest-neighbor spacing distribution is also long-tailed (not Wigner-Dyson) and, after proper scaling, very close to the result for the 2 x 2 case -- a generalization of Wigner's surmise. We discuss connections of these "nonextensive" ensembles with other non-Gaussian ones, like the so-called L\'evy ensembles and those arising from soft-confinement.