Quantum diffusion and delocalization in one-dimensional band matrices via the flow method

We study a class of Gaussian random band matrices of dimension $N\times N$ and band-width $W$. We show that delocalization holds for bulk eigenvectors and that quantum diffusion holds for the resolvent, all under the assumption that $W\gg N^{8\mathrm{/}11}$. Our analysis is based on a flow method, and a refinement of it may lead to an improvement on the condition $W\gg N^{8\mathrm{/}11}$.