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The Scaling Limit of the Directed Polymer with Power-Law Tail Disorder

Authors :
Hubert Lacoin
Quentin Berger
Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001))
Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
Sorbonne Université (SU)
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Instituto Nacional de matematica pura e aplicada
Source :
Communications in Mathematical Physics
Publication Year :
2021
Publisher :
Springer Science and Business Media LLC, 2021.

Abstract

In this paper, we study the so-called intermediate disorder regime for a directed polymer in a random environment with heavy-tail. Consider a simple symmetric random walk $(S_n)_{n\geq 0}$ on $\mathbb{Z}^d$, with $d\geq 1$, and modify its law using Gibbs weights in the product form $\prod_{n=1}^{N} (1+\beta\eta_{n,S_n})$, where $(\eta_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a field of i.i.d. random variables whose distribution satisfies $\mathbb{P}(\eta>z) \sim z^{-\alpha}$ as $z\to\infty$, for some $\alpha\in(0,2)$. We prove that if $\alpha< \min(1+\frac{2}{d},2)$, when sending $N$ to infinity and rescaling the disorder intensity by taking $\beta=\beta_N \sim N^{-\gamma}$ with $\gamma =\frac{d}{2\alpha}(1+\frac{2}{d}-\alpha)$, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with L\'evy $\alpha$-stable noise constructed in the companion paper arXiv:2007.06484.<br />Comment: 48 pages, comments are welcome

Details

ISSN :
14320916 and 00103616
Volume :
386
Database :
OpenAIRE
Journal :
Communications in Mathematical Physics
Accession number :
edsair.doi.dedup.....136cb9c84f0fd1d71171e4c90d0859ba