Overlaps and Pathwise Localization in the Anderson Polymer Model

TL;DR

This paper investigates the long-term behavior of paths in the Anderson polymer model, revealing how the measure concentrates around typical paths and exhibits localization as parameters grow large, using a pathwise approach.

Contribution

It introduces a pathwise analysis of polymer localization outside the perturbative regime, utilizing overlap estimates and Gaussian integration by parts.

Findings

Polymer measure concentrates near typical paths as time grows.

Localization becomes complete as 2/7 approaches infinity.

The canonical measure exhibits scaling, thermodynamic limits, and parameter decoupling.

Abstract

We consider large time behavior of typical paths under the Anderson polymer measure. If $P$ is the measure induced by rate $\kappa,$ simple, symmetric random walk on $Z^d$ started at $x,$ this measure is defined as $$ d\mu(X)={Z^{-1} \exp\{\beta\int_0^T dW_{X(s)}(s)\}dP(X) $$ where $\{W_x:x\in Z^d\}$ is a field of $iid$ standard, one-dimensional Brownian motions, $\beta>0, \kappa>0$ and $Z$ the normalizing constant. We establish that the polymer measure gives a macroscopic mass to a small neighborhood of a typical path as $T \to \infty$, for parameter values outside the perturbative regime of the random walk, giving a pathwise approach to polymer localization, in contrast with existing results. The localization becomes complete as $\frac{\beta^2}{\kappa}\to\infty$ in the sense that the mass grows to 1. The proof makes use of the overlap between two independent samples drawn under the Gibbs measure $\mu$, which can be estimated by the integration by parts formula for the Gaussian environment. Conditioning this measure on the number of jumps, we obtain a canonical measure which already shows scaling properties, thermodynamic limits, and decoupling of the parameters.