LOCAL LAW AND TRACY–WIDOM LIMIT FOR SPARSE SAMPLE COVARIANCE MATRICES

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 limiti...

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Bibliographic Details
Published in:The Annals of applied probability 2019-10, Vol.29 (5), p.3006-3036
Main Authors: Hwang, Jong Yun, Lee, Ji Oon, Schnelli, Kevin
Format: Article
Language:English
Subjects:
Covariance matrix
Eigenvalues
Graph theory
High-dimensional sample covariance matrices
local law
Matrix
Probability
Spectra
Tracy-Widom distribution
Variation
Vertex sets
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Summary: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 Erd˝os–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.
ISSN:1050-5164
2168-8737
2168-8737
DOI:10.1214/19-AAP1472