Non-directed polymers in heavy-tail random environment in dimension $d\geq 2$

Abstract

In this article we study a \emph{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 $(\omega_x)_{x\in \mathbb{Z}^d}$. The model consists in modifying the law of the random walk up to time (or length) $N$ by the exponential of $\sum_{x\in \mathcal{R}_N}\beta (\omega_x-h)$ where $\mathcal{R}_N$ is the range of the walk, \textit{i.e.} the set of visited sites up to time $N$, and $\beta\geq 0,\, h\in \mathbb{R}$ are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking $\beta:=\beta_N$ vanishing as the length $N$ goes to infinity, and in the case where the random variables $\omega$ have a heavy tail with exponent $\alpha\in (0,d)$. We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes $\beta_N = \hat \beta N^{-\gamma}$ with $\gamma \geq 0$: we find the correct transversal fluctuation exponent $\xi$ for the polymer (it depends on $\alpha$ and $\gamma$) 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.