Overlaps and Pathwise Localization in the Anderson Polymer Model

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.

Data and Resources

Additional Info

Field Value
Source https://hal.science/hal-00607809
Author Comets, Francis, Cranston, Michael
Maintainer CCSD
Last Updated May 28, 2026, 23:00 (UTC)
Created May 28, 2026, 23:00 (UTC)
Identifier hal-00607809
Language en
Rights https://about.hal.science/hal-authorisation-v1/
contributor Laboratoire de Probabilités et Modèles Aléatoires (LPMA) ; Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
creator Comets, Francis
date 2012-12-20T00:00:00
harvest_object_id 9d170af9-9aad-46ee-8a2d-bf0681b186f3
harvest_source_id 3374d638-d20b-4672-ba96-a23232d55657
harvest_source_title test moissonnage SELUNE
metadata_modified 2025-09-29T00:00:00
relation info:eu-repo/semantics/altIdentifier/arxiv/1107.2011
set_spec type:UNDEFINED