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We consider Greens functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(mathbb Z/Lmathbb Z)^d$. The entries $h_{xy}=overline h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=mathbb E|h_{xy}|^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any $nin mathbb N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 sum_{alpha}s_{xalpha}|G_{alpha y}|^2$, with an error $O(W^{-nd/2})$, and use it to prove a local law on the Greens function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $dge 8$ in part I of this series.
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
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
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
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