2*2 random matrix ensembles with reduced symmetry: From Hermitian to PT-symmetric matrices

TL;DR

This paper introduces new 2x2 random matrix ensembles with reduced symmetry, exploring their spectral statistics and extending classical ensembles to include PT-symmetric matrices with novel properties.

Contribution

It proposes symmetry-constrained random matrix ensembles, including PT-symmetric cases, and analyzes their spectral statistics, revealing new distributions and connections to classical ensembles.

Findings

Novel level-spacing distributions such as singular and half-Gaussian distributions.

PT-symmetric ensembles with U(2) invariance exhibit GUE or truncated-GUE statistics.

Symmetry reduction leads to new insights into spectral properties of random matrices.

Abstract

A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or parity-time- (PT-) symmetric matrices. To illustrate the main idea, we first study 2*2 complex Hermitian matrix ensembles with O(2) invariant constraints, yielding novel level-spacing statistics such as singular distributions, half-Gaussian distribution, distributions interpolating between GOE (Gaussian Orthogonal Ensemble) distribution and half Gaussian distributions, as well as gapped-GOE distribution. Such a symmetry-reduction strategy is then used to explore 2*2 PT-symmetric matrix ensembles with real eigenvalues. In particular, PT-symmetric random matrix ensembles with U(2) invariance can be constructed, with the conventional complex Hermitian random matrix ensemble being a special case. In two examples of PT-symmetric random matrix ensembles, the level-spacing distributions are found to be the standard GUE (Gaussian Unitary Ensemble) statistics or "truncated-GUE" statistics.