On Symmetries of the Feinberg-Zee Random Hopping Matrix

TL;DR

This paper investigates the spectral properties of the Feinberg-Zee random hopping matrix, revealing explicit polynomial symmetries and their implications for the structure of its spectrum, including connections to Julia sets.

Contribution

The paper explicitly characterizes a class of polynomial symmetries of the matrix's spectrum, including Chebyshev-related polynomials, and explores their impact on spectral set topology.

Findings

Identifies a class of polynomials preserving the spectrum set.

Shows the spectrum's interior is dense and contains filled Julia sets.

Establishes connections between spectral symmetries and complex dynamics.

Abstract

In this paper we study the spectrum $\Sigma$ of the infinite Feinberg-Zee random hopping matrix, a tridiagonal matrix with zeros on the main diagonal and random $\pm 1$'s on the first sub- and super-diagonals; the study of this non-selfadjoint random matrix was initiated in Feinberg and Zee (Phys. Rev. E 59 (1999), 6433--6443). Recently Hagger (arXiv:1412.1937, Random Matrices: Theory Appl.}, {\bf 4} 1550016 (2015)) has shown that the so-called periodic part $\Sigma_\pi$ of $\Sigma$, conjectured to be the whole of $\Sigma$ and known to include the unit disk, satisfies $p^{-1}(\Sigma_\pi) \subset \Sigma_\pi$ for an infinite class $S$ of monic polynomials $p$. In this paper we make very explicit the membership of $S$, in particular showing that it includes $P_m(\lambda) = \lambda U_{m-1}(\lambda/2)$, for $m\geq 2$, where $U_n(x)$ is the Chebychev polynomial of the second kind of degree $n$. We also explore implications of these inverse polynomial mappings, for example showing that $\Sigma_\pi$ is the closure of its interior, and contains the filled Julia sets of infinitely many $p\in S$, including those of $P_m$, this partially answering a conjecture of the second author.