A Block Reduction Method for Random Band Matrices with General Variance Profiles

Abstract: We present a novel block reduction method for the study of a general class of random band matrices (RBM) defined on the $d$-dimensional lattice $\mathbb{Z}_{L}^d:={1,2,\ldots,L}^{d}$ for $d\in {1,2}$, with band width $W$ and an almost arbitrary variance profile subject to a core condition. We prove the delocalization of bulk eigenvectors for such RBMs under the assumptions $W\ge L^{1/2+\varepsilon}$ in one dimension and $W\geq L^{\varepsilon}$ in two dimensions, where $\varepsilon$ is an arbitrarily small constant. This result extends the findings of arXiv:2501.01718 and arXiv:2503.07606 on block RBMs to models with general variance profiles. Furthermore, we generalize our results to Wegner orbital models with small interaction strength $\lambda\ll 1$. Under the sharp condition $\lambda\gg W^{-d/2}$, we establish optimal lower bounds for the localization lengths of bulk eigenvectors, thereby extending the results of arXiv:2503.11382 to settings with nearly arbitrary potential and hopping terms. Our block reduction method provides a powerful and flexible framework that reduces both the dynamical analysis of the loop hierarchy and the derivation of deterministic estimates for general RBMs to the corresponding analysis of block RBMs, as developed in arXiv:2501.01718, arXiv:2503.07606 and arXiv:2503.11382.