# Overlaps and Pathwise Localization in the Anderson Polymer Model

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.
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Preprints, Working Papers, ...
2012
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Cited literature [31 references]

https://hal.archives-ouvertes.fr/hal-00607809
Contributor : Francis Comets <>
Submitted on : Thursday, December 20, 2012 - 1:56:13 PM
Last modification on : Thursday, April 27, 2017 - 9:45:55 AM
Document(s) archivé(s) le : Thursday, March 21, 2013 - 3:49:13 AM

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• HAL Id : hal-00607809, version 3
• ARXIV : 1107.2011

### Citation

Francis Comets, Michael Cranston. Overlaps and Pathwise Localization in the Anderson Polymer Model. 2012. 〈hal-00607809v3〉

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