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This is the second part of a three part series of papers. In this paper, we consider a general class of $N\times N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $\mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1\ll W\ll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=\left(z\mathds{1}_{1\le i \le W}+\wt z\mathds{1}_{ i > W} \right) \delta_{ij}$, with two distinct spectral parameters $z\in \mathbb C_+:=\{z\in \mathbb C:\im z>0\}$ and $\wt z\in \mathbb C_+\cup \mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $W\gg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in [2]. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in [10].

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No abstract uploaded!

We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.