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Directed polymers in a random environment with heavy tails
Author(s) -
Auffinger Antonio,
Louidor Oren
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20348
Subject(s) - scaling , mathematics , diagonal , statistical physics , limit (mathematics) , polymer , limiting , lipschitz continuity , entropy (arrow of time) , mathematical analysis , physics , geometry , thermodynamics , mechanical engineering , nuclear magnetic resonance , engineering
We study the model of directed polymers in a random environment in 1 + 1 dimensions, where the distribution at a site has a tail that decays regularly polynomially with power α, where α ∈ (0,2). After proper scaling of temperature β −1 , we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (α, β)‐indexed family of measures on Lipschitz curves lying inside the 45°‐rotated square with unit diagonal. In particular, this shows order‐ n transversal fluctuations of the polymer. If, and only if, α is small enough, we find that there exists a random critical temperature below which, but not above which, the effect of the environment is macroscopic. The results carry over to d + 1 dimensions for d > 1 with minor modifications. © 2010 Wiley Periodicals, Inc.