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 @math equals @math or @math (the diffusive phase is reduced to @math ) while when @math , there is a critical temperature @math which delimits the two phases. The proof of the existence of a diffusive regime for @math 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 @math ), has finite second moment. The aim of this work is to investigate how the presence absence of phase transition may depend on the dimension @math in the case when the disorder variable displays heavier tail. To this end we replace @math by @math where @math is in the domain of attraction of a stable law with parameter @math .
Among other works that deal with heavy tail environments we mention @cite_13 , where in the setup the environment @math is allowed to belong to the domain of attraction of a @math -stable law and it is studied properties of paths trajectories drawn from the polymer measure. In this context there is no free energy so the work is fundamentally different from ours. In @cite_10 , it is studied the influence of the jump distribution on the delocalization-localization transition and the interplay between jump tails, spatial dimension and existence of the delocalized phase, when nearest neighbor walks are replaced by long range jumps. Our results most likely extend to that setup, the criterion for having a weak disorder phase in dimension @math becoming @math where @math is the exponent of the random walk. We also mention @cite_3 as another case where a change in the environment setup (there is: moving from the IID setup to a strong spatial correlation in the environment) modifies the criterion for having no phase transition.
@cite_10 @cite_13 @cite_3
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In this paper, we consider directed polymers in random environment with long range jumps in discrete space and time. We extend to this case some techniques, results and classifications known in the usual short range case. However, some properties are drastically different when the underlying random walk belongs to the domain of attraction of an @math -stable law. For instance, we construct natural examples of directed polymers in random environment which experience weak disorder in low dimension. We study the directed polymer model in dimension @math when the environment is heavy-tailed, with a decay exponent @math . We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature @math vanishes as the size of the system n goes to infinity. When @math , we show that all possible transversal fluctuations @math can be achieved by tuning properly @math , allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when @math , we show that there are only two regimes: the transversal fluctuations are either @math or @math . As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP].