A class of 2x2 correlated random-matrix models with Brody spacing distribution

TL;DR

This paper introduces a new class of 2x2 random-matrix models with exact Brody distribution eigenvalue spacing, differing from classical ensembles by relaxing independence, Gaussianity, and hermiticity constraints, allowing for diverse spectral statistics.

Contribution

The paper presents a novel 2x2 random-matrix model class with exact Brody spacing distribution, extending spectral analysis beyond traditional Hermitian and Gaussian assumptions.

Findings

Brody distribution is the exact eigenvalue spacing for the new models.

Models include non-Hermitian matrices with real or complex eigenvalues.

Transitions between different eigenvalue spacing statistics are demonstrated.

Abstract

A class of 2x2 random-matrix models is introduced for which the Brody distribution is the exact eigenvalue spacing distribution. The matrix elements consist of constrained finite sums of an exponential random variable raised to various powers that depend on the Brody parameter. The random matrices introduced here differ from those of the Gaussian Orthogonal Ensemble (GOE) in three important ways: the matrix elements are not independent and identically distributed (i.e., not IID) nor Gaussian-distributed, and the matrices are not necessarily real and/or symmetric. The first two features arise from dropping the classical independence assumption, and the third feature stems from dropping the quantum-mechanical conditions imposed in the construction of the GOE. In particular, the hermiticity condition, which in the present class of models, is a sufficient but not necessary condition for the eigenvalues to be real, is not imposed. Consequently, complex non-Hermitian 2x2 random matrices with real or complex eigenvalues can also have spacing distributions that are intermediate between those of the Poisson and Wigner classes. Numerical examples are provided for different types of random matrices, including complex-symmetric matrices with real or complex-conjugate eigenvalues. Various generalizations and extensions are discussed including a simple modification that effectuates cross-over transitions between other classes of eigenvalue spacing statistics. The case of a cross-over transition between semi-Poisson and Ginibre spacing statistics is presented as a novel example.