## The Annals of Probability

- Ann. Probab.
- Volume 44, Number 6 (2016), 4006-4048.

### High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder

Partha S. Dey and Nikos Zygouras

#### Abstract

The directed polymer model at intermediate disorder regime was introduced by Alberts–Khanin–Quastel [*Ann. Probab.* **42** (2014) 1212–1256]. It was proved that at inverse temperature $\beta n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $\gamma$.

#### Article information

**Source**

Ann. Probab., Volume 44, Number 6 (2016), 4006-4048.

**Dates**

Received: April 2015

Revised: September 2015

First available in Project Euclid: 14 November 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1479114269

**Digital Object Identifier**

doi:10.1214/15-AOP1067

**Mathematical Reviews number (MathSciNet)**

MR3572330

**Zentralblatt MATH identifier**

1359.60117

**Subjects**

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

Secondary: 60G57: Random measures 60G70: Extreme value theory; extremal processes

**Keywords**

Directed polymer phase transition heavy tail scaling limits

#### Citation

Dey, Partha S.; Zygouras, Nikos. High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder. Ann. Probab. 44 (2016), no. 6, 4006--4048. doi:10.1214/15-AOP1067. https://projecteuclid.org/euclid.aop/1479114269