One-arm exponent of critical level-set for metric graph Gaussian free field in high dimensions

Abstract

In this paper, we study the critical level-set of Gaussian free field (GFF) on the metric graph $\widetilde{\mathbb{Z}}^d,d>6$. We prove that the one-arm probability (i.e. the probability of the event that the origin is connected to the boundary of the box $B(N)$) is proportional to $N^{-2}$, where $B(N)$ is centered at the origin and has side length $2\lfloor N \rfloor$. Our proof is hugely inspired by Kozma and Nachmias [29] which proves the analogous result of the critical bond percolation for $d\geq 11$, and by Werner [51] which conjectures the similarity between the GFF level-set and the bond percolation in general and proves this connection for various geometric aspects.