Delocalization for a class of random block band matrices

We consider $N\times N$ Hermitian random matrices $H$ consisting of blocks of size $M\geq N^{6/7}$. 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 Green's 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 Green's function comparison strategy. Previous estimates were valid only for $\eta\gg M^{-1}$. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.