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

This paper is concerned with the union sp Ω ( j , k ) ⁡ T n ( a ) sp_Ω (j,k) T_n(a) of all possible spectra that may emerge when perturbing a large n × n n × 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 Ω . 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 ) sp_Ω (j,k) T_n(a) as n → ∞ n → ∞ . Also discussed are the cases of small and large sets Ω Ω as well as the “discontinuity of the infinite volume case”, which means that in general sp Ω ( j , k ) ⁡