Directed polymers in heavy-tail random environment

Quentin Berger (1) , Torri Niccolo (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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 $\sqrt{n}\le h_n\le n$ 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. On the other hand, when $\alpha<1/2$, we show that there are only two regimes: the transversal fluctuations are either $\sqrt{n}$ or $n$. As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP].

Mots clés: Directed polymer; Heavy-tail distributions; Intermediate disorder; Last Passage Percolation; Super-diffusivity; Weak-coupling limit

Domaines: Mathématiques [math]; Probabilités [math.PR]; Physique mathématique [math-ph]

Quentin Berger, Torri Niccolo. Directed polymers in heavy-tail random environment. Annals of Probability, inPress. ⟨hal-01706666v2⟩