Random band matrices in the delocalized phase, III: Averaging fluctuations

We consider a general class of symmetric or Hermitian random band matrices H=(hxy)x,y ∈ 1,Nᵈ in any dimension d≥ 1, where the entries are independent, centered random variables with variances sxy= E|hxy|². We assume that sxy vanishes if $|x-y|$ exceeds the band width W, and we are interested in the mesoscopic scale with 1 W N. Define the {generalized resolvent} of H as G(H,Z):=(H - Z)⁻¹, where Z is a deterministic diagonal matrix with entries Zₓₓ∈ C₊ for all x. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions d≥ 2. More precisely, for any fixed d≥ 2, we prove that the bulk eigenvectors of H are delocalized in certain averaged sense if N≤ W1+d/2. This improves the corresponding results in {HeMa2018} under the assumption N W1+d/d+1, and in {ErdKno2013,ErdKno2011} under the assumption N W1+d/6. For 1D random band matrices, our fluctuation averaging result was used in {PartII,PartI} to prove the delocalization conjecture and bulk universality for random band matrices with N W4/3.