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.References
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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