Anderson localisation for quasionedimensional 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 multiscale 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 multiscale analysis, and also extend it to the general quasionedimensional 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 singlescale proofs of localisation in pure one dimension ($W=1$), recently found by BucajDamanikFillmanGerbuzVandenBoomWangZhang, JitomirskayaZhu, GorodetskiKleptsyn, and Rangamani.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2110.00097
 Bibcode:
 2021arXiv211000097M
 Keywords:

 Mathematical Physics;
 Mathematics  Probability;
 Mathematics  Spectral Theory
 EPrint:
 17pp