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General Toeplitz Matrices Subject to Gaussian Perturbations

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Abstract

We study the spectra of general \(N\times N\) Toeplitz matrices given by symbols in the Wiener Algebra perturbed by small complex Gaussian random matrices, in the regime \(N\gg 1\). We prove an asymptotic formula for the number of eigenvalues of the perturbed matrix in smooth domains. We show that these eigenvalues follow a Weyl law with probability sub-exponentially close to 1, as \(N\gg 1\), in particular that most eigenvalues of the perturbed Toeplitz matrix are close to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.

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Notes

  1. Here \({{\mathcal {N}}}(A)\) and \({{\mathcal {R}}}(A)\) denote the null space and the range of a linear operator A.

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Acknowledgements

The first author acknowledges support from the 2018 S. Bergman award. The second author was supported by a CNRS Momentum fellowship. We are grateful to Ofer Zeitouni for his interest and a remark which lead to a better presentation of this paper. We are grateful to the referee for pointing out a mistake affecting the range of the exponent \(\delta _1\).

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Authors and Affiliations

  1. IMB, UMR 5584 du CNRS, Université de Bourgogne Franche-Comté, 9, avenue Alain Savary, BP 47870 , 21078, Dijon Cedex, France

    Johannes Sjöstrand

  2. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000, Strasbourg, France

    Martin Vogel

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  1. Johannes Sjöstrand
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  2. Martin Vogel
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Correspondence to Martin Vogel.

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Communicated by Jan Derezinski.

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Sjöstrand, J., Vogel, M. General Toeplitz Matrices Subject to Gaussian Perturbations. Ann. Henri Poincaré 22, 49–81 (2021). https://doi.org/10.1007/s00023-020-00970-w

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