Overlaps and pathwise localization in the Anderson polymer model

We consider large time behaviour of typical paths under the Anderson polymer measure. If P κ x is the measure induced by rate κ , simple, symmetric random walk on Z d started at x , this measure is defined as d μ κ , β , T x ( X ) = Z κ , β , T ( x ) − 1 exp { β ∫ 0 T d W X ( s ) ( s ) } d P κ x ( X ) where { W x : x ∈ Z d } is a field of i i d standard, one-dimensional Brownian motions, β > 0 , κ > 0 and Z κ , β , T ( x ) the normalizing constant. We establish that the polymer measure gives a macroscopic mass to a small neighbourhood of a typical path as T → ∞ , 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 β 2 κ → ∞ 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 κ , β , T x , 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.