Toeplitz band matrices with small random perturbations

Johannes Sjöstrand (1) , Martin Vogel (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IRMA - Institut de Recherche Mathématique Avancée

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.

Mots clés: Spectral theory; Non-self-adjoint operators; Random perturbations

Domaines: Théorie spectrale [math.SP]; Probabilités [math.PR]

Johannes Sjöstrand, Martin Vogel. Toeplitz band matrices with small random perturbations. Indagationes Mathematicae, 2021, 32 (1), pp.275-322. ⟨10.1016/j.indag.2020.09.001⟩. ⟨hal-02975964⟩