Spectral approximation of banded Laurent matrices with localized random perturbations

Abstract

This paper explores the relationship between the spectra of perturbed infinite banded Laurent matrices $L(a)+K$ and their approximations by perturbed circulant matrices $C_{n}(a)+P_{n}KP_{n}$ for large $n$. The entries $K_{jk}$ of the perturbation matrices assume values in prescribed sets $\Omega_{jk}$ at the sites $(j,k)$ of a fixed set $E$, and are zero at the sites $(j,k)$ outside $E$. With ${\cal K}_{\Omega}^{E}$ denoting the ensemble of these perturbation matrices, it is shown that \ud \displaystyle\lim_{n\to\infty} \ud \displaystyle\bigcup_{K\in{\cal K}_{\Omega}^{E}}\ud sp(C_{n}(a)+P_{n}KP_{n})=\ud \displaystyle\bigcup_{K\in{\cal K}_{\Omega}^{E}}\ud sp(L(a)=K)\ud under several fairly general assumptions on $E$ and $\Omega$