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 $d\ge 2$: we consider a simple symmetric random walk on ${\mathbb{Z}}^d$ which interacts with a random environment, represented by i.i.d. random variables ${({\mathit{\omega }}_x)}_{x\in {\mathbb{Z}}^d}$. The model consists in modifying the law of the random walk up to time (or length) N by the exponential of ${\sum }_{x\in {\mathcal{R}}_N}\mathit{\beta }({\mathit{\omega }}_x-h)$ where ${\mathcal{R}}_N$ is the range of the walk, i.e. the set of visited sites up to time N, and $\mathit{\beta }\ge 0,h\in \mathbb{R}$ are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking $\mathit{\beta }:={\mathit{\beta }}_N$ vanishing as the length N goes to infinity, and in the case where the random variables ω have a heavy tail with exponent $\mathit{\alpha }\in (0,d)$. We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes ${\mathit{\beta }}_N=\hat{\mathit{\beta }}{N}^{-\mathit{\gamma }}$ with $\mathit{\gamma }\ge 0$: 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.