Anderson localisation for quasi-one-dimensional random operators

TL;DR

This paper provides a new proof of Anderson localisation for quasi-one-dimensional random operators, extending previous results by avoiding multi-scale analysis and establishing exponential decay properties.

Contribution

It offers a novel proof technique for Anderson localisation in quasi-one-dimensional models, including cases with random hopping, and derives sharp bounds on eigenfunction correlators.

Findings

Proved Anderson localisation without multi-scale analysis.

Extended localisation results to models with random hopping.

Established exponential decay of eigenfunctions and Fermi projections.

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. 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.