The local semicircle law for a general class of random matrices

Erdős, László and Knowles, Antti and Yau, Horng-Tzer and Yin, Jun (2013) The local semicircle law for a general class of random matrices. Electronic Journal of Probability, 18. Article No. 59. ISSN 1083-6489

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We consider a general class of N × N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij|2. As a consequence, we prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W ≫N1-εn with some εn > 0 and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17, 19, 6].

Item Type: Article
DOI: 10.1214/EJP.v18-2473
Uncontrolled Keywords: Eigenvalue rigidity, Local semicircle law, Random band matrix, Universality
Subjects: 500 Science > 530 Physics
Research Group: Erdös Group
SWORD Depositor: Sword Import User
Depositing User: Sword Import User
Date Deposited: 04 Jan 2016 14:38
Last Modified: 05 Sep 2017 14:22

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