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Toeplitz band matrices with small random perturbations

Abstract : We study the spectra of N × N Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime N 1. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on N , with probability sub-exponentially (in N) close to 1. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most O(N −1+ε), for all ε > 0, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.
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Submitted on : Friday, October 23, 2020 - 10:29:12 AM
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Johannes Sjöstrand, Martin Vogel. Toeplitz band matrices with small random perturbations. Indagationes Mathematicae, Elsevier, In press, ⟨10.1016/j.indag.2020.09.001⟩. ⟨hal-02975964⟩

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