A classical problem of statistical mechanics is to understand the behaviour of a polymer in interaction with a solvent containing impurities. From a mathematical point of view, the polymer is described by a random walk on $\mathbb{Z}^d$ of length $N$ placed in a time-independent random environment (the impurities).
Over the last forty years most of the efforts in this area of research in mathematics and theoretical physics have focused on the analysis of the directed polymer model, that is a model which describes a polymer stretched along a given direction. Based on the recent articles with Q. Berger, C.-H. Huang and R. Wei (arXiv:2101.05949, arXiv:2002.06899), in this talk we present a non-directed polymer model: we consider a simple symmetric random walk on $\mathbb{Z}^d, \, d\ge 1$ where the interaction with the environment occurs on the range of the random walk, i.e. on the set of sites visited by the walk. This model can also be viewed as a randomly perturbed version of random walks penalized or rewarded by their ranges.
We discuss the results obtained for this model under the assumption that the environment has heavy-tail, that is the distribution function of the environment decades polynomially with an exponent $\alpha>0$, and we compare them to the results already known in the directed polymer framework.
