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October 2019 Local law and Tracy–Widom limit for sparse sample covariance matrices
Jong Yun Hwang, Ji Oon Lee, Kevin Schnelli
Ann. Appl. Probab. 29(5): 3006-3036 (October 2019). DOI: 10.1214/19-AAP1472

Abstract

We consider spectral properties of sparse sample covariance matrices, which includes biadjacency matrices of the bipartite Erdős–Rényi graph model. We prove a local law for the eigenvalue density up to the upper spectral edge. Under a suitable condition on the sparsity, we also prove that the limiting distribution of the rescaled, shifted extremal eigenvalues is given by the GOE Tracy–Widom law with an explicit formula on the deterministic shift of the spectral edge. For the biadjacency matrix of an Erdős–Rényi graph with two vertex sets of comparable sizes $M$ and $N$, this establishes Tracy–Widom fluctuations of the second largest eigenvalue when the connection probability $p$ is much larger than $N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.

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Jong Yun Hwang. Ji Oon Lee. Kevin Schnelli. "Local law and Tracy–Widom limit for sparse sample covariance matrices." Ann. Appl. Probab. 29 (5) 3006 - 3036, October 2019. https://doi.org/10.1214/19-AAP1472

Information

Received: 1 August 2018; Revised: 1 December 2018; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155065
MathSciNet: MR4019881
Digital Object Identifier: 10.1214/19-AAP1472

Subjects:
Primary: 60B20, 62H10

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2019
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