The Local Semicircle Law for a General Class of Random Matrices

2012/12/01 by Erdos, Laszlo, Knowles, Antti, Yau, Horng-Tzer +1 · 5 citations
#15B52 #82B44 #82C44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1212.0164

Abstract

We consider a general class of N× 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 [14] 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 \E \absh_ij². 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≫ N^1-\e_n with some \e_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 from [3,4,16].

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