We consider a directed polymer model in dimension , where the random walk is attracted to stable law and the environment is independent in time variable and correlated in space variable. We obtain the scaling limit in the intermediate disorder regime for partition function, and show that the rescaled point-to-point partition function of directed polymers converges in the space of continuous functions to the solution of a stochastic heat equation driven by time-white spatial-colored noise. The scaling limit of the polymer transition probability is also established in the path space. The proof of the tightness is based on the gradient estimates for symmetric random walks in the domain of normal attraction of α-stable law which are established in this paper.
Supported by NSFC Grant 11971361 and 11731012.
The authors are very grateful to the anonymous referees for the careful reading and many helpful comments and suggestions that greatly improved the paper. The authors also wish to thank Professor Guanglin Rang for his useful discussions.
"Scaling limits of directed polymers in spatial-correlated environment." Electron. J. Probab. 28 1 - 57, 2023. https://doi.org/10.1214/23-EJP955