Directed polymers in heavy-tail random environment

Understand

We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $\alpha\in(0,2)$.

• We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature $\beta=\beta_n$ vanishes as the size of the system $n$ goes to infinity.
• When $\alpha\in(1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n} \leq h_n \leq n$ can be achieved by tuning properly $\beta_n$, 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.