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

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We consider spectral properties of sparse sample covariance matrices, which includes biadjacency matrices of the bipartite Erdos-Renyi 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 Erdos-Renyi 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).
Publisher
INST MATHEMATICAL STATISTICS
Issue Date
2019-10
Language
English
Article Type
Article
Citation

ANNALS OF APPLIED PROBABILITY, v.29, no.5, pp.3006 - 3036

ISSN
1050-5164
DOI
10.1214/19-AAP1472
URI
http://hdl.handle.net/10203/268206
Appears in Collection
MA-Journal Papers(저널논문)
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