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

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 $2q$ 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.

14 pages including 5 figures

References in corpus (4)

• Random-Matrix Theory of Quantum Transport
• Random Matrix Theories in Quantum Physics: Common Concepts
• Distribution of Eigenvalues in Non-Hermitian Anderson Model
• Density of states in the non-hermitian Lloyd model