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 $κ,$ simple, symmetric random walk on $Z^d$ started at $x,$ this measure is defined as $$ dμ(X)={Z^{-1} \exp\{β\int_0^T dW_{X(s)}(s)\}dP(X) $$ where $\{W_x:x\in Z^d\}$ is a field of $iid$ standard, one-dimensional Brownian motions, $β>0, κ>0$ 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 $\frac{β^2}κ\to\infty$ 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 $μ$, 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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