Diffusion Profile for Random Band Matrices: A Short Proof

Let H be a Hermitian random matrix whose entries H_xy are independent, centred random variables with variances S_xy = E|H_xy|², where x, y ∈ (Z/LZ)^d and d ⩾ 1. The variance S_xy is negligible if |x − y| is bigger than the band width W. For d =1 we prove that if L ≪ W^1+2/7 then the eigenvectors of H are delocalized and that an averaged version of |G_xy (z)|² exhibits a diffusive behaviour, where G (z) = (H − z)⁻¹ is the resolvent of H. This improves the previous assumption L ≪ W^1+1/4 of Erdős et al. (Commun Math Phys 323:367–416, 2013). In higher dimensions d ⩾ 2, we obtain similar results that improve the corresponding ones from Erdős et al. (Commun Math Phys 323:367–416, 2013). Our results hold for general variance profiles S_xy and distributions of the entries H_xy. The proof is considerably simpler and shorter than that of Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013), Erdős et al. (Commun Math Phys 323:367–416, 2013). It relies on a detailed Fourier space analysis combined with isotropic estimates for the fluctuating error terms. It is completely self-contained and avoids the intricate fluctuation averaging machinery from Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013).