The local semicircle law for a general class of random matrices

László Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin
2013 Electronic Journal of Probability  
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. Abstract 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
more » ... 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−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] . expect the universality results of [17, 18, 19 ] to hold. In addition to a transition in the local spectral statistics, an accompanying transition is conjectured to occur in the behaviour localization length of the eigenvectors of H, whereby in the large-W regime they are expected to be completely delocalized and in the small-W regime exponentially localized. The localization length for band matrices was recently investigated in great detail in [8] . EJP 18 (2013), paper 59. Page 2/58 s ij C/M (1.3) on the variances (instead of (1.1)). Here M is a new parameter that typically satisfies M N . (From now on, the relation A B for two N -dependent quantities A and B means that A N −ε B for some positive ε > 0.) The question of the validity of the local semicircle law under the assumption (1.3) was initiated in [17], where (1.2) was proved with an error term of order (M η) −1/2 away from the spectral edges. The purpose of this paper is twofold. First, we prove a local semicircle law (1.2), under the variance condition (1.3), with a stronger error bound of order (M η) −1 , including energies E near the spectral edge. Away from the spectral edge (and from the origin E = 0 if the matrix does not have a band structure), the result holds for any η 1/M . Near the edge there is a restriction on how small η can be. This restriction EJP 18 (2013), paper 59.
doi:10.1214/ejp.v18-2473 fatcat:tgfzqpo7xfcvdnuixn4wtkjoze