Accessibility navigation

On symmetries of the Feinberg-Zee random hopping matrix

Chandler-Wilde, S. N. and Hagger, R. (2017) On symmetries of the Feinberg-Zee random hopping matrix. In: Maz'ya, V., Natroshvili, D., Shargorodsky, E. and Wendland, W. L. (eds.) Recent Trends in Operator Theory and Partial Differential Equations: the Roland Duduchava Anniversary Volume. Operator Theory: Advances and Applications, 258. Birkhauser, Basel. ISBN 9783319470771

Full text not archived in this repository.

It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing.


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 ({\it Phys. Rev. E} {\bf 59} (1999), 6433--6443). Recently Hagger ({\em Random Matrices: Theory Appl.}, {\bf 4} 1550016 (2015)) has shown that the so-called {\em 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 $\cS$ of monic polynomials $p$. In this paper we make very explicit the membership of $\cS$, 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 $p\in \cS$, including those of $P_m$, this partially answering a conjecture of the second author.

Item Type:Book or Report Section
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:67214

University Staff: Request a correction | Centaur Editors: Update this record

Page navigation