  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.
