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 \(z\rightarrow \infty \), 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 (Berger and Lacoin in The continuum directed polymer in Lévy Noise, 2020. arXiv:2007.06484v2).

在本文中，我们研究了具有重尾的随机环境中定向聚合物的所谓中间无序机制。考虑在\（\ mathbb {Z} ^ d \）上使用\（d \ ge 1 \）的简单对称随机游动\（（S_n）_ {n \ ge 0} \），并使用Gi​​bbs权重修改其定律以产品形式\（\ prod _ {n = 1} ^ {N}（1+ \ beta \ eta _ {n，S_n}）\），其中\（（（eta _ {n，x}）_ { n \ ge 0，x \ in {\ mathbb {Z}} ^ d} \）是iid随机变量的字段，其分布满足\（{\ mathbb {P}}（\ eta> z）\ sim z ^ { -\ alpha} \）为\（z \ rightarrow \ infty \），对于某些\（\ alpha \ in（0,2）\）。我们证明如果\（\ alpha <\ min（1 + \ frac {2} {d}，2）\），当将N发送到无穷大并通过将\（\ beta = \ beta _N \ sim N ^ {-\ gamma} \）与\（\ gamma = \ frac {d} {2 \ alpha}（1 + \ frac {2} {d}-\ alpha）\）时，扩散标度下的轨迹分布在法律上趋向于一个随机极限，这是由Lévy \（\ alpha \） -稳定噪声构成的连续谱聚合物随行论文（Berger和Lacoin在LévyNoise的“连续体定向聚合物”中，2020年。arXiv：2007.06484v2）。