10.1214/EJP.v18-2473
Erdös, László
László
Erdös0000-0001-5366-9603
Knowles, Antti
Antti
Knowles
Yau, Horng
Horng
Yau
Yin, Jun
Jun
Yin
The local semicircle law for a general class of random matrices
Institute of Mathematical Statistics
2013
2018-12-11T11:59:51Z
2019-08-02T12:37:53Z
journal_article
https://research-explorer.app.ist.ac.at/record/2837
https://research-explorer.app.ist.ac.at/record/2837.json
651497 bytes
application/pdf
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].