## The Annals of Applied Probability

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

#### 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 $(Np)^{-1}$.

#### Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1571385628

Digital Object Identifier
doi:10.1214/19-AAP1472

Mathematical Reviews number (MathSciNet)
MR4019881

#### Citation

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. https://projecteuclid.org/euclid.aoap/1571385628

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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.