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We consider the eigenvalues of a fixed, non-normal matrix subject to a smalladditive perturbation. In particular, we consider the case when the fixedmatrix is a banded Toeplitz matrix, where the bandwidth is allowed to growslowly with the dimension, and the perturbation matrix is drawn from one ofseveral different random matrix ensembles. We establish a number ofnon-asymptotic results for the eigenvalues of this model, including a local lawand a rate of convergence in Wasserstein distance of the empirical spectralmeasure to its limiting distribution. In addition, we define the classicallocations of the eigenvalues and prove a rigidity result showing that, onaverage, the eigenvalues concentrate closely around their classical locations.While proving these results we also establish a number of auxiliary resultsthat may be of independent interest, including a quantitative version of theTao--Vu replacement principle, a general least singular value bound thatapplies to adversarial models, and a description of the limiting empiricalspectral measure for random multiplicative perturbations.

Quantitative results for banded Toeplitz matrices subject to random and deterministic perturbations