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:

14322064

Publisher:

Springer

Date Published:

20170614

Start Page: 
Epub ahead of print

Sponsor: 
Samsung Science and Technology Foundation project number SSTFBA140204; ERC Advanced Grant RANMAT No. 338804; Göran Gustafsson Foundation

URL: 

DOI: 
10.1007/s0044001707878

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) 