Non-directed polymers in random environments with range penalties: the high-dimensional case

We study a non-directed polymer model in random environments.The polymer is modeled by a simple symmetric random walk S on Z d with d ¥ 2, and the random environment is modeled by i.i.d.random variables whose tail probability decays polynomially.The interaction between the polymer and the random environment is captured by a Gibbs transform: at time N , the law of S is tilted by the factor expp ° xPR N pβω x hqq, where R N is the range of S up to time N , β ¥ 0 is the inverse temperature, and h P R is an external field.By appropriately tuning β β N and h h N , we establish the phase diagram, analyze the fluctuations of S under the Gibbs transform, and derive the scaling limits of the (logarithmic) partition function.This paper is a follow-up work of Berger et al. (2022b).The main novelty and challenge arise from tuning the external field h, which brings in various range penalties, unlike in Berger et al. (2022b), where h is fixed and serves only as a centering term for the random environment.