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REGULARIZATION OF NON-NORMAL MATRICES BY GAUSSIAN NOISE—THE BANDED TOEPLITZ AND TWISTED TOEPLITZ CASES

Part of: Probability theory on algebraic and topological structures Special matrices

Published online by Cambridge University Press:  13 February 2019

ANIRBAN BASAK Open the ORCID record for ANIRBAN BASAK [Opens in a new window] ,
ELLIOT PAQUETTE Open the ORCID record for ELLIOT PAQUETTE [Opens in a new window]  and
OFER ZEITOUNI Open the ORCID record for OFER ZEITOUNI [Opens in a new window]
ANIRBAN BASAK
Affiliation:
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India; anirban.basak@icts.res.in 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 W 18th Ave, Columbus, Ohio 43210, USA; paquette.30@osu.edu
OFER ZEITOUNI
Affiliation:
Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel Courant Institute, New York University, 251 Mercer St, New York, NY 10012, USA; ofer.zeitouni@weizmann.ac.il

Abstract

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We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_{N}$ be a deterministic $N\times N$ matrix, and let $G_{N}$ be a complex Ginibre matrix. We consider the matrix ${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where $\unicode[STIX]{x1D6FE}>1/2$. With $L_{N}$ the empirical measure of eigenvalues of ${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties $L_{N}$ to the singular values of $z-M_{N}$, with $z\in \mathbb{C}$. We then compute the limit of $L_{N}$ when $M_{N}$ is an upper-triangular Toeplitz matrix of finite symbol: if $M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$ where $\mathfrak{d}$ is fixed, $a_{i}\in \mathbb{C}$ are deterministic scalars and $J$ is the nilpotent matrix $J(i,j)=\mathbf{1}_{j=i+1}$, then $L_{N}$ converges, as $N\rightarrow \infty$, to the law of $\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$ where $U$ is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in $M_{N}$.

MSC classification

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see )
Secondary: 15B05: Toeplitz, Cauchy, and related matrices 15B52: Random matrices
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e3
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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