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The local semicircle law for a general class of random matrices


 
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1. Title Title of document The local semicircle law for a general class of random matrices
 
2. Creator Author's name, affiliation, country László Erdős; LMU-University of Munich
 
2. Creator Author's name, affiliation, country Antti Knowles; New York University; United States
 
2. Creator Author's name, affiliation, country Horng-Tzer Yau; Harvard University; United States
 
2. Creator Author's name, affiliation, country Jun Yin; University of Wisconsin; United States
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random band matrix; local semicircle law; universality; eigenvalue rigidity
 
3. Subject Subject classification 15B52; 82B44
 
4. Description Abstract We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \mathbb{E} \left|h_{ij}\right|^2$. 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  $W\gg N^{1-\varepsilon_n}$ with some $\varepsilon_n>0$ 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSF, DFG
 
7. Date (YYYY-MM-DD) 2013-05-29
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2473
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-2473
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
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