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

Abstract

We study a non-directed polymer model in random environments. The polymer is modeled by a simple symmetric random walk $S$ on ${\mathbb{Z}}^d$ with $d\ge 2$, 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 $\exp ({\sum }_{x\in {\mathcal{R}}_N}({\hat{I}}^2\ddot{I}{‰}_x-h))$, where ${\mathcal{R}}_N$ is the range of $S$ up to time $N$, ${\hat{I}}^2\ge 0$ is the inverse temperature, and $h\in \mathbb{R}$ is an external field. By appropriately tuning ${\hat{I}}^2={\hat{I}}_N^2$ 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 arxiv.org/abs/2101.05949. The main novelty and challenge arise from tuning the external field $h$, which brings in various range penalties, unlike in arxiv.org/abs/2101.05949, where $h$ is fixed and serves only as a centering term for the random environment.

28 pages, 2 figures