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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 block to form a block-band structure, with an essential band width $M$. We show that the entries of the Greens function $G(z)=(H-z)^{-1}$ satisfy the local semicircle law with spectral parameter $z=E+mathbf{i}eta$ down to the real axis for any $eta gg N^{-1}$, using a combination of the supersymmetry method inspired by cite{Sh2014} and the Greens function comparison strategy. Previous estimates were valid only for $etagg M^{-1}$. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.
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 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
We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(mathbb Z/Lmathbb Z)^d$. The entries $h_{xy}$ are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances $s_{xy}=mathb
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
We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.