Overlaps and pathwise localization in the Anderson polymer model

We consider large time behaviour of typical paths under the Anderson polymer measure. If Pκx is the measure induced by rate κκ, simple, symmetric random walk on ZdZd started at xx, this measure is defined as dμκ,β,Tx(X)=Zκ,β,T(x)−1exp{β∫0TdWX(s)(s)}dPκx(X) where {Wx:x∈Zd}{Wx:x∈Zd} is a field of iidiid standard, one-dimensional Brownian motions, β>0,κ>0β>0,κ>0 and Zκ,β,T(x)Zκ,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→∞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 β2κ→∞ 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 μκ,β,Tx, 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.