论文标题
带有幂律尾部疾病的定向聚合物的缩放限制
The scaling limit of the directed polymer with power-law tail disorder
论文作者
论文摘要
在本文中，我们研究了带有重尾的随机环境中的定向聚合物的所谓中间疾病制度。考虑一个简单的对称随机步行$（s_n）_ {n \ geq 0} $ on $ \ mathbb {z}^d $，带有$ d \ geq 1 $，并使用gibbs with forders $ \ prod_ prod_ {n = 1}}^n}（1+β{1+βη_} $ prop_ { $（η_{n，x}）_ {n \ ge 0，x \ in \ mathbb {z}^d} $是i.i.d.的一个字段。随机变量的分布满足$ \ mathbb {p}（η> z）\ sim z^{ - α} $作为$ z \ to \ infty $，对于某些$α\ in（0,2）$。我们证明，如果$α<\ min（1+ \ frac {2} {d}，2）$，当将$ n $发送给无穷大并通过$β=β_n\ sim n^{ - γ{ - γ} $进行$β= sim n^{ - γ} $，以$γ= = = \ frac {d} $ frac {2 $ {2} {d} {d} {d}在扩散缩放下的轨迹在法律上会收敛于随机极限，这是伴侣论文arXiv中构建的lévy$α$稳定噪声的连续聚合物：2007.06484。
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 $\prod_{n=1}^{N} (1+βη_{n,S_n})$, where $(η_{n,x})_{n\ge 0, x\in \mathbb{Z}^d}$ is a field of i.i.d. random variables whose distribution satisfies $\mathbb{P}(η>z) \sim z^{-α}$ as $z\to\infty$, for some $α\in(0,2)$. We prove that if $α< \min(1+\frac{2}{d},2)$, when sending $N$ to infinity and rescaling the disorder intensity by taking $β=β_N \sim N^{-γ}$ with $γ=\frac{d}{2α}(1+\frac{2}{d}-α)$, the distribution of the trajectory under diffusive scaling converges in law towards a random limit, which is the continuum polymer with Lévy $α$-stable noise constructed in the companion paper arXiv:2007.06484.
