Anderson Localisation for Quasi-One-Dimensional Random Operators

Abstract

In 1990, Klein, Lacroix, and Speis proved (spectral) Anderson localisation for the Anderson model on the strip of width W geqslant 1, allowing for singular distribution of the potential. Their proof employs multi-scale analysis, in addition to arguments from the theory of random matrix products (the case of regular distributions was handled earlier in the works of Goldsheid and Lacroix by other means). We give a proof of their result avoiding multi-scale analysis and also extend it to the general quasi-one-dimensional model, allowing, in particular, random hopping. Furthermore, we prove a sharp bound on the eigenfunction correlator of the model, which implies exponential dynamical localisation and exponential decay of the Fermi projection. The method is also applicable to operators on the half-line with arbitrary (deterministic, self-adjoint) boundary condition. Our work generalises and complements the single-scale proofs of localisation in pure one dimension (W=1), recently found by Bucaj–Damanik–Fillman–Gerbuz–VandenBoom–Wang–Zhang, Jitomirskaya–Zhu, Gorodetski–Kleptsyn, and Rangamani.