The Localization Scale of the Block Anderson Model in One Dimension at Arbitrary Coupling and Volume

Abstract

We study the one-dimensional block Anderson model with independent $W\times W$ Gaussian unitary ensemble potentials and deterministic nearest-neighbor hopping, with hopping-to-disorder ratio $g$. The system consists of $L$ blocks, with total dimension $N=WL$. As $W\to\infty$, we identify the bulk localization length, up to an arbitrarily small power of $W$, with $\min\{N,\max\{W,W^2g^2\}\}$, uniformly for deterministic couplings $g\ge0$ and cycle lengths $L\ge3$. For every fixed $D>0$ and all sufficiently large $W$, the expected fraction of eigenvectors in the selected energy interval satisfying these bounds is at least $1-W^{-D}$. In polynomial volume the bounds hold simultaneously for all eigenvectors in that interval with probability at least $1-W^{-D}$. The localization length is the number of coordinates in the shortest cyclic interval containing at least half of an eigenvector's mass.