Local law and Tracy–Widom limit for sparse random matrices Journal Article

Author(s): Lee, Jii O; Schnelli, Kevin
Article Title: Local law and Tracy–Widom limit for sparse random matrices
Affiliation IST Austria
Abstract: 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.
Keywords: Local law; sparse random matrices; Erdős–Rényi graph
Journal Title: Probability Theory and Related Fields
ISSN: 1432-2064
Publisher: Springer  
Date Published: 2017-06-14
Start Page: Epub ahead of print
Sponsor: Samsung Science and Technology Foundation project number SSTF-BA1402-04; ERC Advanced Grant RANMAT No. 338804; Göran Gustafsson Foundation
DOI: 10.1007/s00440-017-0787-8
Notes: We thank László Erdös for useful comments and suggestions. Ji Oon Lee is grateful to the department of mathematics, University of Michigan, Ann Arbor, for their kind hospitality during the academic year 2014–2015
Open access: yes (repository)