Localization Lengths of Power-Law Random Band Matrices

We study large $N\times N$ power-law random band matrices $H=(H_{ij})$ with centered complex Gaussian entries, where the variances satisfy a power-law decay $\mathbb{E}|H_{ij}|^2\propto (|i-j|/W+1)^{-1-α}$, for some exponent $α>-1$ and bandwidth $W\gg 1$. We establish the following lower bounds, wit