On Symmetries of the Feinberg–Zee Random Hopping Matrix

Abstract

In this paper we study the spectrum Σ of the infinite Feinberg–Zee random hopping matrix, a tridiagonal matrix with zeros on the main diagonal and random ±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 (Random Matrices: Theory Appl., 4 1550016 (2015)) has shown that the so-called periodic part Σ_π of Σ, conjectured to be the whole of Σ 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 $P_m(\lambda)\;=\;\lambda U_{m-1},\;\mathrm{for}\;m\;\geq 2$, where Un(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 Σ_π is the closure of its interior, and contains the filled Julia sets of infinitely many $p\in\;\mathcal{S}$, including those of P _m, this partially answering a conjecture of the second author.

Mathematics Subject Classification (2010). Primary 47B80; Secondary 37F10, 47A10, 47B36, 65F15.

Dedicated to Roland Duduchava on the occasion of his 70th birthday