## Annals of Probability

### Directed polymers in heavy-tail random environment

#### Abstract

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, that is, 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 superdiffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [Ann. Probab. 44 (2016) 4006–4048]—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 [Ann. Appl. Probab. 29 (2019) 1878–1903].

#### Article information

Source
Ann. Probab., Volume 47, Number 6 (2019), 4024-4076.

Dates
Revised: January 2019
First available in Project Euclid: 2 December 2019

https://projecteuclid.org/euclid.aop/1575277347

Digital Object Identifier
doi:10.1214/19-AOP1353

Mathematical Reviews number (MathSciNet)
MR4038048

Zentralblatt MATH identifier
07212177

#### Citation

Berger, Quentin; Torri, Niccolò. Directed polymers in heavy-tail random environment. Ann. Probab. 47 (2019), no. 6, 4024--4076. doi:10.1214/19-AOP1353. https://projecteuclid.org/euclid.aop/1575277347

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