Random cyclic matrices

TL;DR

This paper introduces a Gaussian ensemble of real cyclic matrices, analyzes their spectral properties, and finds that their eigenvalue spacings follow distributions similar to those in well-known random matrix models, with applications across physics.

Contribution

The paper provides the first analytical and numerical study of spectral fluctuations in Gaussian ensembles of real cyclic matrices, revealing their pseudo-symmetric nature and eigenvalue spacing distributions.

Findings

Eigenvalue spacings follow Gaussian or linear distributions for small spacings.

Spacing distribution matches Wigner distribution for Poisson process on a plane.

Results applicable to disordered systems, statistical mechanics, and quantum chromodynamics.

Abstract

We present a Gaussian ensemble of random cyclic matrices on the real field and study their spectral fluctuations. These cyclic matrices are shown to be pseudo-symmetric with respect to generalized parity. We calculate the joint probability distribution function of eigenvalues and the spacing distributions analytically and numerically. For small spacings, the level spacing distribution exhibits either a Gaussian or a linear form. Furthermore, for the general case of two arbitrary complex eigenvalues, leaving out the spacings among real eigenvalues, and, among complex conjugate pairs, we find that the spacing distribution agrees completely with the Wigner distribution for Poisson process on a plane. The cyclic matrices occur in a wide variety of physical situations, including disordered linear atomic chains and Ising model in two dimensions. These exact results are also relevant to two-dimensional statistical mechanics and $\nu$-parametrized quantum chromodynamics.