Non-Hermitian tridiagonal random matrices and returns to the origin of a random walk

TL;DR

This paper investigates a class of non-Hermitian tridiagonal random matrices, revealing their eigenvalue distributions form symmetric polygons and connecting their spectral properties to counting specific closed random walks.

Contribution

It introduces the 'q-roots of unity' matrix models, analyzes their eigenvalue densities, and links matrix traces to combinatorial walk counting, providing explicit evaluations.

Findings

Eigenvalue densities form regular polygons in the complex plane.

Averaged traces of matrix powers count specific closed random walks.

Explicit formulas for walk counts are derived.

Abstract

We study a class of tridiagonal matrix models, the "q-roots of unity" models, which includes the sign ($q=2$) and the clock ($q=\infty$) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with $2 q$ sides, in the complex plane. Furthermore the averaged traces of $M^k$ are integers that count closed random walks on the line, such that each site is visited a number of times multiple of $q$. We obtain an explicit evaluation for them.