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Matrix regularizing effects of Gaussian perturbations

Author(s): Aizenman, Michael; Peled, Ron; Schenker, Jeffrey; Shamis, Mira; Sodin, Sasha

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Abstract: The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for H = A + V, where A is the base matrix and V is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of H in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of H-1 and for the distribution of the norm of H-1 applied to a fixed vector. The bounds are uniform in A and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.
Publication Date: Jun-2017
Electronic Publication Date: 9-Mar-2017
Citation: Aizenman, Michael, Peled, Ron, Schenker, Jeffrey, Shamis, Mira, Sodin, Sasha. (2017). Matrix regularizing effects of Gaussian perturbations. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 19 (10.1142/S0219199717500286
DOI: doi:10.1142/S0219199717500286
ISSN: 0219-1997
EISSN: 1793-6683
Type of Material: Journal Article
Version: Author's manuscript

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