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July 2021 Eigenvector statistics of Lévy matrices
Amol Aggarwal, Patrick Lopatto, Jake Marcinek
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Ann. Probab. 49(4): 1778-1846 (July 2021). DOI: 10.1214/20-AOP1493

Abstract

We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Lévy matrices) whose associated eigenvalues are sufficiently small. We show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue. Although the latter random variable is typically nonexplicit, for the median eigenvector it is given by the inverse of a one-sided stable law. Moreover, we show that different entries of the same eigenvector are asymptotically independent, but that there are nontrivial correlations between eigenvectors with nearby eigenvalues. Our findings contrast sharply with the known eigenvector behavior for Wigner matrices and sparse random graphs.

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Amol Aggarwal. Patrick Lopatto. Jake Marcinek. "Eigenvector statistics of Lévy matrices." Ann. Probab. 49 (4) 1778 - 1846, July 2021. https://doi.org/10.1214/20-AOP1493

Information

Received: 1 March 2020; Revised: 1 November 2020; Published: July 2021
First available in Project Euclid: 13 May 2021

Digital Object Identifier: 10.1214/20-AOP1493

Subjects:
Primary: 15B52, 60B20

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 4 • July 2021
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