## Abstract

In this article we study a non-directed polymer model in dimension d ≥ 2: we consider a simple symmetric random walk on Z^{d} which interacts with a random environment, represented by i.i.d. random variables (ω_{x})_{x}_{ϵZ}d. The model consists in modifying the law of the random walk up to time (or length) N by the exponential of (Formula presented) where R_{N} is the range of the walk, i.e. the set of visited sites up to time N, and β ≥ 0, h ϵ R are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking β:= β_{N} vanishing as the length N goes to infinity, and in the case where the random variables ω have a heavy tail with exponent α ϵ (0, d). We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes (Formula presented) with γ ≥ 0: we find the correct transversal fluctuation exponent ξ for the polymer (it depends on α and γ) and we give the limiting distribution of the rescaled log-partition function. This extends existing works to the non-directed case and to higher dimensions.

Original language | English |
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Article number | 148 |

Journal | Electronic Journal of Probability |

Volume | 27 |

DOIs | |

Publication status | Published - 2022 |

Externally published | Yes |

## Keywords

- heavy-tail distributions
- random polymer
- random walk
- range
- sub-diffusivity
- super-diffusivity
- weak-coupling limit