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In the present work, we investigate the case of Directed Polymer in a RandomEnvironment (DPRE), when the increments of the random walk are heavy-tailedwith tail-exponent equal to zero ($\mathbf{P}[|X_1|\geq n]$ decays slower thanany power of $n$). This case has not yet been studied in the context ofdirected polymers and present key differences with the simple symmetric randomwalk case and the cases where the increments belong to the domain of attractionof an $\alpha$-stable law, where $\alpha \in (0, 2]$. We establish the absenceof a very strong disorder regime - that is, the free energy equals zero atevery temperature - for every disorder distribution. We also prove that astrong disorder regime (partition function converging to zero at lowtemperature) may exist or not depending on finer properties of the random walk:we establish non-matching necessary and sufficient conditions for having aphase transition from weak to strong disorder. In particular our results implythat for this directed polymer model, very strong disorder is not equivalent tostrong disorder, shedding a new light on a long standing conjecture concerningthe original nearest-neighbor DPRE.

Directed Polymer for very heavy tailed random walks