Localization for a nonlinear sigma model in a strip related to vertex reinforced jump processes

TL;DR

This paper proves exponential localization in a lattice sigma model related to Anderson localization and vertex reinforced jump processes, using a Mermin-Wagner argument and transfer operator approach.

Contribution

It establishes exponential localization for a 3D lattice sigma model in a strip, connecting localization phenomena with vertex reinforced jump processes.

Findings

Proves exponential localization at any temperature in a strip.

Extends localization results to quasi-one dimensional graphs with pinning.

Uses Mermin-Wagner type argument and transfer operator method.

Abstract

We study a lattice sigma model which is expected to reflect Anderson localization and delocalization transition for real symmetric band matrices in 3D, but describes the mixing measure for a vertex reinforced jump process too. For this model we prove exponential localization at any temperature in a strip, and more generally in any quasi-one dimensional graph, with pinning (mass) at only one site. The proof uses a Mermin-Wagner type argument and a transfer operator approach.