Translator Disclaimer
November 2019 Directed polymers in heavy-tail random environment
Quentin Berger, Niccolò Torri
Ann. Probab. 47(6): 4024-4076 (November 2019). DOI: 10.1214/19-AOP1353

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].

Citation

Download Citation

Quentin Berger. Niccolò Torri. "Directed polymers in heavy-tail random environment." Ann. Probab. 47 (6) 4024 - 4076, November 2019. https://doi.org/10.1214/19-AOP1353

Information

Received: 1 May 2018; Revised: 1 January 2019; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212177
MathSciNet: MR4038048
Digital Object Identifier: 10.1214/19-AOP1353

Subjects:
Primary: 60F05, 82D60
Secondary: 60G70, 60K37

Rights: Copyright © 2019 Institute of Mathematical Statistics

JOURNAL ARTICLE
53 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.47 • No. 6 • November 2019
Back to Top