Random reverse-cyclic matrices and screened harmonic oscillator

TL;DR

This paper derives the joint probability distribution for random reverse-cyclic matrices, linking it to an exactly solvable N-body model called a screened harmonic oscillator, revealing unique eigenvalue density and spacing distributions.

Contribution

It introduces a novel connection between reverse-cyclic matrices and a solvable N-body model, providing new insights into eigenvalue distributions and correlations.

Findings

Eigenvalue density follows a Wigner form with a hole at the origin.

Spacing distributions vary from Gaussian-like to Wigner.

Connection to an exactly solvable N-body model.

Abstract

We have calculated the joint probability distribution function for random reverse-cyclic matrices and shown that it is related to an N-body exactly solvable model. We refer to this well-known model potential as a screened harmonic oscillator. The connection enables us to obtain all the correlations among the particle positions moving in a screened harmonic potential. The density of nontrivial eigenvalues of this ensemble is found to be of the Wigner form and admits a hole at the origin, in contrast to the semicircle law of the Gaussian orthogonal ensemble of random matrices. The spacing distributions assume different forms ranging from Gaussian-like to Wigner.