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

In this paper, we study the so-called intermediate disorder regime for a directed polymer in random environment with heavy-tail. Consider simple symmetric walk $$(S_n)_{n\ge 0}$$ on $$\mathbb {Z}^d$$ , $$d\ge 1$$ and modify its law using Gibbs weights 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 field of i.i.d. variables whose distribution satisfies $${\mathbb {P}}(\eta >z) \sim z^{-\alpha }$$ as $$z\rightarrow \infty $$ some $$\alpha \in (0,2)$$ . We prove that if < \min (1 + \frac{2}{d} ,2)$$ when sending N to infinity rescaling intensity by taking $$\beta =\beta _N N^{-\gamma $$\gamma =\frac{d}{2\alpha }(1+\frac{2}{d}-\alpha )$$ trajectory under diffusive scaling converges towards limit, which continuum Lévy -stable noise constructed companion paper (Berger Lacoin The Noise, 2020. arXiv:2007.06484v2 ).