The Annals of Applied Probability

Local law and Tracy–Widom limit for sparse sample covariance matrices

Jong Yun Hwang, Ji Oon Lee, and Kevin Schnelli

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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 $(Np)^{-1}$.

Article information

Ann. Appl. Probab., Volume 29, Number 5 (2019), 3006-3036.

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

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Mathematical Reviews number (MathSciNet)

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 62H10: Distribution of statistics

High-dimensional sample covariance matrices local law Tracy–Widom distribution


Hwang, Jong Yun; Lee, Ji Oon; Schnelli, Kevin. Local law and Tracy–Widom limit for sparse sample covariance matrices. Ann. Appl. Probab. 29 (2019), no. 5, 3006--3036. doi:10.1214/19-AAP1472.

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Supplemental materials

  • Supplement: Proofs of some lemmas. In the Supplementary Material [18], we will provide the proofs of Lemmas 4.1, 4.2, 4.4 and 5.4.