Directed polymers in a random environment with heavy tails

Antonio Auffinger*, Oren Louidor

*Corresponding author for this work

Research output: Contribution to journalArticle

10 Scopus citations


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 ε (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.

Original languageEnglish (US)
Pages (from-to)183-204
Number of pages22
JournalCommunications on Pure and Applied Mathematics
Issue number2
StatePublished - Feb 1 2011

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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