DIRECTED POLYMERS IN HEAVY-TAIL RANDOM ENVIRONMENT AND ENTROPY-CONTROLLED LAST PASSAGE PERCOLATION

Quentin Berger (1) , Torri Niccolo (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 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 \emph{weak-coupling} regime, \textit{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 $\sqrt{n} \leq h_n \leq n$ of the polymer 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 \cite{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 $n$. This extends the results of Auffinger and Louidor \cite{AL11}, and Dey and Zygouras \cite{cf:DZ}, which considered only the cases where $h_n =n$, resp.~$h_n=\sqrt{n}$. As a key ingredient, we introduce the \emph{Entropy-controlled Last Passage Percolation} (E-LPP), which is a natural generalization of Hammersley's Last Passage Percolation where points can be collected by paths with the constraint to have an entropy bounded by a fixed constant -- instead of a $1$-Lipschitz constraint. We prove several estimates on the E-LPP in continuous and in discrete settings, which are of interest on their own.

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

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

Quentin Berger, Torri Niccolo. DIRECTED POLYMERS IN HEAVY-TAIL RANDOM ENVIRONMENT AND ENTROPY-CONTROLLED LAST PASSAGE PERCOLATION. 2018. ⟨hal-01706666v1⟩