Abstract
In this article we study a non-directed polymer model in dimension : we consider a simple symmetric random walk on which interacts with a random environment, represented by i.i.d. random variables . The model consists in modifying the law of the random walk up to time (or length) N by the exponential of where is the range of the walk, i.e. the set of visited sites up to time N, and are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking vanishing as the length N goes to infinity, and in the case where the random variables ω have a heavy tail with exponent . We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes with : 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.
Funding Statement
Q. Berger, N. Torri and R. Wei are supported by a public grant overseen by the French National Research Agency, ANR SWiWS (ANR-17-CE40-0032-02). N. Torri was also supported by the Labex MME-DII.
Acknowledgments
The authors would like to thank the referees for their comments and suggestions, which helped improve the quality of the article.
Citation
Quentin Berger. Niccolò Torri. Ran Wei. "Non-directed polymers in heavy-tail random environment in dimension ." Electron. J. Probab. 27 1 - 67, 2022. https://doi.org/10.1214/22-EJP873
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