Delocalization and Quantum Diffusion of Random Band Matrices in High Dimensions II: T -expansion

We consider the Green’s function G(z):=(H-z)^-1 of Hermitian random band matrix H on the d -dimensional lattice (ℤ/Lℤ)^d . The entries h_xy=h_yx of H are independent centered complex Gaussian random variables with variances s_xy=𝔼|h_xy|^2 , which satisfy a banded profile so that s_xy is negligible if |x-y| exceeds the band width W . For any fixed n∈ℕ , we construct an expansion of the T -variable, T_xy=|m|^2 ∑ _αs_xα|G_α y|^2 , with an error O (W^-nd/2) , and use it to prove a local law on the Green’s function. This T -expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions d⩾ 8 in part I (Yang et al. in Delocalization and quantum diffusion of random band matrices in high dimensions I: self-energy renormalization, 2021. arXiv:2104.12048 ) of this series.