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Transactions of the American Mathematical Society

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Spectrum of random perturbations of Toeplitz matrices with finite symbols


Authors: Anirban Basak, Elliot Paquette and Ofer Zeitouni
Journal: Trans. Amer. Math. Soc. 373 (2020), 4999-5023
MSC (2010): Primary 60B20
DOI: https://doi.org/10.1090/tran/8040
Published electronically: March 3, 2020
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Abstract: Let $ T_N$ denote an $ N\times N$ Toeplitz matrix with finite, $ N$ independent symbol $ \bm {a}$. For $ E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that $ {N^{-1/2}\Vert E_N\Vert _{{\mathrm {HS}}}}\to _{N\to \infty } 0$ at a polynomial rate), we prove that the empirical measure of eigenvalues of $ T_N+E_N$ converges to the law of $ \bm {a}(U)$, where $ U$ is uniformly distributed on the unit circle in the complex plane. This extends results from [Forum Math. Sigma 7 (2019)] to the non-triangular setup and non-complex Gaussian noise, and confirms predictions obtained in [Linear Algebra Appl. 162/164 (1992), pp. 153-185] using the notion of pseudospectrum.


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Additional Information

Anirban Basak
Affiliation: International Center for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India; and Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel

Elliot Paquette
Affiliation: Department of Mathematics, The Ohio State University, Tower 100, 231 West 18th Avenue, Columbus, Ohio 43210

Ofer Zeitouni
Affiliation: Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel; and Courant Institute, New York University, 251 Mercer Street, New York, New York 10012

DOI: https://doi.org/10.1090/tran/8040
Received by editor(s): December 15, 2018
Received by editor(s) in revised form: November 8, 2019
Published electronically: March 3, 2020
Additional Notes: The first author was partially supported by a Start-up Research Grant (SRG/2019/001376) from Science and Engineering Research Board of Govt. of India, and ICTS–Infosys Excellence Grant.
The third author was partially supported by Israel Science Foundation grant 147/15 and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement number 692452).
Article copyright: © Copyright 2020 American Mathematical Society