The Annals of Probability

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

Partha S. Dey and Nikos Zygouras

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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


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