Spectrum of a Feinberg-Zee Random Hopping Matrix

TL;DR

This paper proves that the spectra of certain infinite random matrices almost surely include the unit disc and extends the result to matrices with a spectral hole, requiring significant analytical modifications.

Contribution

It offers a new proof for the spectral properties of a class of infinite random matrices and generalizes to matrices with a spectral hole around the origin.

Findings

Spectra contain the unit disc almost surely.

Spectra of more general matrices include a hole around zero.

Analytical methods adapted for matrices with spectral holes.

Abstract

This paper provides a new proof of a theorem of Chandler-Wilde, Chonchaiya and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra contain a hole around the origin. The presence of the hole forces substantial changes to the analysis.