Delocalization for a class of random block band matrices

Abstract

We consider N× N Hermitian random matrices H consisting of blocks of size M≥ N6 / 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+ iη down to the real axis for any η≫ N- 1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫ M- 1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.