Spectrum of random perturbations of Toeplitz matrices with finite symbols
| Autor: | Elliot Paquette, Anirban Basak, Ofer Zeitouni |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Polynomial (hyperelastic model)
Applied Mathematics General Mathematics Probability (math.PR) 010102 general mathematics Spectrum (functional analysis) 01 natural sciences Toeplitz matrix Complex normal distribution Combinatorics Matrix (mathematics) Unit circle FOS: Mathematics 0101 mathematics Complex plane Mathematics - Probability Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Transactions of the American math. Society |
| ISSN: | 1088-6850 0002-9947 0091-1798 0021-9045 1073-7928 0002-9939 0025-5521 0219-1997 0024-3795 0894-0347 |
| Popis: | Let $T_N$ denote an $N\times N$ Toeplitz matrix with finite, $N$ independent symbol ${\bf a}$. For $E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that $N^{-1/2}\|E_N\|_{{\rm HS}}\to_{N\to\infty} 0$ at a polynomial rate), we prove that the empirical measure of eigenvalues of $T_N+E_N$ converges to the law of ${\bf a}(U)$, where $U$ is uniformly distributed on the unit circle in the complex plane. This extends results from arXiv:1712.00042 to the non-triangular setup and non complex Gaussian noise, and confirms predictions obtained in Reichel and Trefethen (1992) using the notion of pseudo-spectrum. Comment: 20 pages, to appear in Trans. Amer. Math. Soc |
| Databáze: | OpenAIRE |
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