The scaling limit of the directed polymer with power-law tail disorder

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 $(S_n{)}_{n\ge 0}$ on ${\mathbb{Z}}^d$, with $d\ge 1$, 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 Lévy $\alpha$-stable noise constructed in the companion paper arXiv:2007.06484.

48 pages, comments are welcome

Cited by in corpus (2)

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