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We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random band matrices and the problem of their Anderson transition. We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.
We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $Wsim N$. All previous results concerning universality of no
We consider $Ntimes N$ Hermitian random matrices $H$ consisting of blocks of size $Mgeq N^{6/7}$. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to blo
This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $Ntimes N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry
Consider $Ntimes N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W geq N^{3/4+varepsilon}$ for any $varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the foll
Recently, T. and M. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint