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2022 Noise sensitivity for the top eigenvector of a sparse random matrix
Charles Bordenave, Jaehun Lee
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Electron. J. Probab. 27: 1-50 (2022). DOI: 10.1214/22-EJP770


We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let v be the top eigenvector of an N×N sparse random symmetric matrix with an average of d non-zero centered entries per row. We resample k randomly chosen entries of the matrix and obtain another realization of the random matrix with top eigenvector v[k]. Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove that, if dN29, with high probability, when kN53, the vectors v and v[k] are almost collinear and, on the contrary, when kN53, the vectors v and v[k] are almost orthogonal. A similar result holds for the eigenvector associated to the second largest eigenvalue of the adjacency matrix of an Erdős-Rényi random graph with average degree dN29.

Funding Statement

CB was supported by the research grant ANR-16-CE40-0024-01. JL was supported by the National Research Foundation of Korea (NRF-2017R1A2B2001952; NRF-2019R1A5A1028324).


We thank the referees for their careful reading of the manuscript and many helpful suggestions.


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Charles Bordenave. Jaehun Lee. "Noise sensitivity for the top eigenvector of a sparse random matrix." Electron. J. Probab. 27 1 - 50, 2022.


Received: 18 June 2021; Accepted: 29 March 2022; Published: 2022
First available in Project Euclid: 27 April 2022

Digital Object Identifier: 10.1214/22-EJP770

Primary: 60B20

Keywords: Noise sensitivity , sparse random matrix


Vol.27 • 2022
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