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

Abstract

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