## Annals of Probability

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

### Directed polymers in heavy-tail random environment

Quentin Berger and Niccolò Torri

#### 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**

Received: May 2018

Revised: January 2019

First available in Project Euclid: 2 December 2019

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/19-AOP1353

**Mathematical Reviews number (MathSciNet)**

MR4038048

**Zentralblatt MATH identifier**

07212177

**Subjects**

Primary: 60F05: Central limit and other weak theorems 82D60: Polymers

Secondary: 60K37: Processes in random environments 60G70: Extreme value theory; extremal processes

**Keywords**

Directed polymer heavy-tail distributions weak-coupling limit last-passage percolation superdiffusivity

#### 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