### Abstract

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 language | English (US) |
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Pages (from-to) | 183-204 |

Number of pages | 22 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 64 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2011 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications on Pure and Applied Mathematics*,

*64*(2), 183-204. https://doi.org/10.1002/cpa.20348