DIRECTED POLYMERS IN HEAVY-TAIL RANDOM ENVIRONMENT

• 成果类型: Article
• 署名作者: Berger, Quentin; Torri, Niccolo
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1353
• 发表日期: 2019
• 页码: 4024-4076
• 关键词: exponents; LIMITS

摘要

We study the directed polymer model in dimension 1 + 1 when the environment is heavy-tailed, with a decay exponent alpha is an element of (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 is an element of (1/2, 2), we show that all possible transversal fluctuations root n <= h(n) <= 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 root 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].