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

TL;DR

This paper investigates the scaling limits of directed polymers in heavy-tailed random environments, showing convergence to a continuum polymer model driven by Lévy stable noise under intermediate disorder scaling.

Contribution

It establishes the convergence of the directed polymer's trajectory to a Lévy stable noise-driven continuum model in the heavy-tail regime, extending prior results to power-law tail disorders.

Findings

Convergence of the polymer's trajectory to a Lévy stable noise continuum model.

Identification of the precise scaling of disorder intensity for convergence.

Extension of scaling limit results to heavy-tailed disorder distributions.

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.