  The transition from a weak-disorder (diffusive phase) to a strong-disorder
(localized phase) for directed polymers in a random environment is a well
studied phenomenon. In the most common setup, it is established that the phase
transition is trivial when the transversal dimension $d$ equals $1$ or $2$ (the
diffusive phase is reduced to $\beta=0$) while when $d\geq 3$, there is a
critical temperature $\beta_c\in (0,\infty)$ which delimits the two phases. The
proof of the existence of a diffusive regime for $d\geq 3$ is based on a second
moment method, and thus relies heavily on the assumption that the variable
which encodes the disorder intensity (which in most of the mathematics
literature assumes the form $e^{\beta \eta_x}$), has finite second moment. The
aim of this work is to investigate how the presence/absence of phase transition
may depend on the dimension $d$ in the case when the disorder variable displays
heavier tail. To this end we replace $e^{\beta \eta_x}$ by $(1+\beta \omega_x)$
where $\omega_x$ is in the domain of attraction of a stable law with parameter
$\gamma \in (1, 2)$.
