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

Abstract

A possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional Hermitian matrices or paritytime (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, the half-Gaussian distribution, distributions interpolating between the GOE (Gaussian orthogonal ensemble) distribution and half-Gaussian distributions, as well as the 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 the truncated- GUE statistics. © 2012 IOP Publishing Ltd.