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arXiv · 1509.01799

Matrix regularizing effects of Gaussian perturbations

Michael Aizenman·Ron Peled·Jeffrey Schenker·Mira Shamis·Sasha Sodin
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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.

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Michael Aizenman, Ron Peled, Jeffrey Schenker, Mira Shamis, Sasha Sodin. 2015-09-06. Matrix regularizing effects of Gaussian perturbations. https://doi.org/10.1142/s0219199717500286

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