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

• 成果类型: Article
• 署名作者: Hwang, Jong Yun; Lee, Ji Oon; Schnelli, Kevin
• 署名单位: Korea Advanced Institute of Science & Technology (KAIST); Royal Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1472
• 发表日期: 2019
• 页码: 3006-3036
• 关键词: largest eigenvalue; spectral statistics; edge universality; components; number

摘要

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).