Non-directed polymers in heavy-tail random environment in dimension d≥2

Abstract

In this article we study a non-directed polymer model in dimension d≥2: we consider a simple symmetric random walk on Zd which interacts with a random environment, represented by i.i.d. random variables (ωx)x∈Zd. The model consists in modifying the law of the random walk up to time (or length) N by the exponential of ∑x∈RNβ(ωx−h) where RN is the range of the walk, i.e. the set of visited sites up to time N, and β≥0,h∈R are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking β:=βN vanishing as the length N goes to infinity, and in the case where the random variables ω have a heavy tail with exponent α∈(0,d). We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes βN=βˆN−γ with γ≥0: we find the correct transversal fluctuation exponent ξ for the polymer (it depends on α and γ) and we give the limiting distribution of the rescaled log-partition function. This extends existing works to the non-directed case and to higher dimensions.