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 ⩾ 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 ( = 1), recently found by Bucaj-Damanik-Fillman-Gerbuz-VandenBoom-Wang-Zhang, Jitomirskaya-Zhu, Gorodetski-Kleptsyn, and Rangamani.

Non-Lyapunov annealed decay for 1d Anderson eigenfunctions

“…This proof can be extended to quasi-one-dimensional operator, such as the Anderson model on the strip of width W or the more general model studied in Ref. 13. A slightly weaker version of ( 7) is still true in this case (see Ref.…”

Lower Bounds on Anderson-Localised Eigenfunctions on a Strip

“…In [16], a general method to compute the Zariski closure of the group generated by the support of T n (λ) was developed; one of its consequences is that (8) holds for any λ ∈ R also in the generality of ( 2). Now we can state the full result of Klein, Lacroix and Speis [25] (in the current setting, covered by [31]): there is an event of full probability on which each eigenpair…”

“…The general Schrödinger case was settled by Klein, Lacroix and Speis in [25], building on [17]. The argument of [25] can be extended to the general situation (2), once the result of [16] (discussed below) is taken into account; an alternative argument avoiding multi-scale analysis and applicable to the general model (1) (and also to its further generalisation allowing for random hopping) is given in [31]. In this paper, we do not discuss Anderson localisation in dimension d > 1, and refer to the works of Fröhlich and Spencer [9] and Aizenman and Molchanov [2] and also to the monograph of Aizenman and Warzel [3].…”

Dynamical Localization for Random Band Matrices up to $W\ll N^{1/4}$

“…Hermitian) matrices. Such models are known to be completely localized [36,15,49,40] for any W , but the localization length in these studies is not estimated quantitatively.…”