Behavior of the Lattice Gaussian Free Field with Weak Repulsive Potentials

Abstract

We consider the $d\ (\ge 3)$ - dimensional lattice Gaussian free field on $\varLambda _N :=[-N, N]^d\cap \mathbb {Z}^d$ in the presence of a self-potential of the form $U(r)= -b I(|r|\le a)$, $a>0, b\in \mathbb {R}$. When $b>0$, the potential attracts the field to the level around zero and is called square-well pinning. It is known that the field turns to be localized and massive for every $a>0$ and $b>0$. In this paper, we consider the situation that the parameter $b<0$ and self-potentials are imposed on $\varLambda _{\alpha N},\ \alpha \in (0, 1)$. We prove that once we impose this weak repulsive potential from the level $[-a, a]$, most sites are located on the same side and the field is pushed to the same level when the original Gaussian field is conditioned to be positive everywhere, or negative everywhere with probability $\frac{1}{2}$, respectively. This result can be applied to show the similar path behavior for the disordered pinning model in the delocalized regime.