On the one-arm probability for the metric graph Gaussian free field in high dimensions

Abstract

We consider the sign clusters of the Gaussian free field on the cable system of a transient weighted graph with volume growth exponent $α$ and Green function decay exponent $ν$. We prove that the probability that the cluster of a vertex reaches distance $R$ is of order $R^{-β}$ when $2α<3ν$, where $β=α-ν$ is the walk dimension, and that the same holds up to a logarithmic factor in the upper bound when $2α=3ν$. On $\mathbb Z^d$ these conditions read $d>6$ and $d=6$. The upper bound combines a general entropic bound, which compares the law of the cluster under the critical loop soup and under a massive one, with a Simon-Lieb-type inequality for the massive field derived from its Markov property. The arguments are robust and require no transitivity. The lower bound follows from a competition between the capacity and the volume of the cluster. In both bounds, the key quantity is the expected exit time of the random walk from balls, which links the volume of the cluster to its radius.