Matrix regularizing effects of Gaussian perturbations

Michael Aizenman; Ron Peled; Jeffrey Schenker; Mira Shamis; Sasha Sodin

doi:10.1142/S0219199717500286

Matrix regularizing effects of Gaussian perturbations

Michael Aizenman, Ron Peled, Jeffrey Schenker, Mira Shamis, Sasha Sodin

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

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.

Original language	English (US)
Article number	1750028
Journal	Communications in Contemporary Mathematics
Volume	19
Issue number	3
DOIs	https://doi.org/10.1142/S0219199717500286
State	Published - Jun 1 2017

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

Keywords

Gaussian perturbation
Minami estimate
Wegner estimate
deformed GOE
deformed GUE

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10.1142/S0219199717500286

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title = "Matrix regularizing effects of Gaussian perturbations",

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.",

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AU - Aizenman, Michael

AU - Peled, Ron

AU - Schenker, Jeffrey

AU - Shamis, Mira

AU - Sodin, Sasha

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

AB - 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.

KW - Gaussian perturbation

KW - Minami estimate

KW - Wegner estimate

KW - deformed GOE

KW - deformed GUE

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Matrix regularizing effects of Gaussian perturbations

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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