The spectra of large Toeplitz band matrices with a randomly perturbed entry

This paper is concerned with the union $\operatorname{sp}_\Omega^{(j,k)} T_n(a)$of all possible spectra that may emerge when perturbing a large $n \times n$ Toeplitz band matrix $T_n(a)$ in the $(j,k)$ site by a number randomly chosen from some set $\Omega$. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of $\operatorname{sp}_\Omega^{(j,k)} T_n(a)$ as $n \to \infty$. Also discussed are the cases of small and large sets $\Omega$ as well as the ``discontinuity of the infinite volume case'', which means that in general $\operatorname{sp}_\Omega^{(j,k)} T_n(a)$ does not converge to something close to $\operatorname{sp}_\Omega^{(j,k)} T(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case.