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 $\mathbb{E}|H_{ij}|^2\propto (|i-j|/W+1)^{-1-\alpha}$, for some exponent $\alpha>-1$ and bandwidth $W\gg 1$. We establish the following lower bounds, with high probability, on the localization length $\ell$ of bulk eigenvectors in the different regimes of $\alpha$: (1) $\ell=N$ if $-1<\alpha<0$; (2) $\ell \ge W^{C}$ for any large constant $C>0$ if $0 < \alpha <1$; (3) $\ell \ge W^{\alpha/(\alpha-1)}$ if $1 < \alpha <2$; (4) $\ell \ge W^{2}$ if $\alpha > 2$. These results verify the physical conjecture of arXiv:cond-mat/9604163 on the delocalized side. The main difficulty in the proof lies in handling the interplay between the non-mean-field nature of the model and the slow decay of the variance profile. To address this issue, a key technical ingredient is a new dynamical analysis of $T$-variables formed from pairs of resolvent entries of $H$. In contrast to the fundamental works on regular random band matrices with fast-decaying variances in arXiv:2501.01718 and arXiv:2506.06441, this approach does not rely on higher-order resolvent loops.