  We consider the eigenvalues of a fixed, non-normal matrix subject to a small
additive perturbation. In particular, we consider the case when the fixed
matrix is a banded Toeplitz matrix, where the bandwidth is allowed to grow
slowly with the dimension, and the perturbation matrix is drawn from one of
several different random matrix ensembles. We establish a number of
non-asymptotic results for the eigenvalues of this model, including a local law
and a rate of convergence in Wasserstein distance of the empirical spectral
measure to its limiting distribution. In addition, we define the classical
locations of the eigenvalues and prove a rigidity result showing that, on
average, the eigenvalues concentrate closely around their classical locations.
While proving these results we also establish a number of auxiliary results
that may be of independent interest, including a quantitative version of the
Tao--Vu replacement principle, a general least singular value bound that
applies to adversarial models, and a description of the limiting empirical
spectral measure for random multiplicative perturbations.
