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 a random environment with heavy-tail. Consider a simple symmetric random walk ( S n ) n ≥ 0 on Z d , with d ≥ 1 , and modify its law using Gibbs weights in the product form ∏ n = 1 N ( 1 + β η n , S n ) , where ( η n , x ) n ≥ 0 , x ∈ Z d is a field of i.i.d. random variables whose distribution satisfies P ( η > z ) ∼ z - α as z → ∞ , for some α ∈ ( 0 , 2 ) . We prove that if α < min ( 1 + 2 d , 2 ) , when sending N to infinity and rescaling the disorder intensity by taking β = β N ∼ N - γ with γ = d 2 α ( 1 + 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 (Berger and Lacoin in The continuum directed polymer in Lévy Noise, 2020. arXiv:2007.06484v2 ).