Local law and Tracy-Widom limit for sparse random matrices

• 成果类型: Article
• 署名作者: Lee, Ji Oon; Schnelli, Kevin
• 署名单位: Korea Advanced Institute of Science & Technology (KAIST); Royal Institute of Technology; Institute of Science & Technology - Austria
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0787-8
• 发表日期: 2018
• 页码: 543-616
• 关键词: wigner random matrices; semicircle law; eigenvalue statistics; spectral statistics; bulk universality; delocalization; edge; eigenvectors

摘要

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the ErdAs-R,nyi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the ErdAs-R,nyi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue when p is much larger than wth a deterministic shift of order (Np)(-1)..