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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Spectrum of random perturbations of Toeplitz matrices with finite symbols
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by Anirban Basak, Elliot Paquette and Ofer Zeitouni PDF
Trans. Amer. Math. Soc. 373 (2020), 4999-5023 Request permission

Abstract:

Let $T_N$ denote an $N\times N$ Toeplitz matrix with finite, $N$ independent symbol $\mathbfit {a}$. For $E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that ${N^{-1/2}\|E_N\|_{{\mathrm {HS}}}}\to _{N\to \infty } 0$ at a polynomial rate), we prove that the empirical measure of eigenvalues of $T_N+E_N$ converges to the law of $\mathbfit {a}(U)$, 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
  • MR Author ID: 899989
  • Elliot Paquette
  • Affiliation: Department of Mathematics, The Ohio State University, Tower 100, 231 West 18th Avenue, Columbus, Ohio 43210
  • MR Author ID: 868866
  • 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
  • MR Author ID: 186850
  • ORCID: 0000-0002-2520-1525
  • 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).
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 4999-5023
  • MSC (2010): Primary 60B20
  • DOI: https://doi.org/10.1090/tran/8040
  • MathSciNet review: 4127869