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Local law and Tracy-Widom limit for sparse random matrices. (English) Zbl 1429.60012

Summary: We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős-Rényi 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 Erdős-Rényi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue when \(p\) is much larger than \(N^{-2/3}\) with a deterministic shift of order \((Np)^{-1}\).

MSC:

60B20 Random matrices (probabilistic aspects)
62H10 Multivariate distribution of statistics
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