Outliers of perturbations of banded Toeplitz matrices

Abstract

Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance ${\sigma }_n^2/n$ in the asymptotic of high dimension and ${\sigma }_n$ converging to $\sigma \ge 0$. Our results complement and provide new proofs on recent progresses in the case $\sigma =0$. For any $\sigma \ge 0$, we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem.

77 pages