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

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 highly inspired by Kozma and Nachmias (J Am Math Soc 24(2):375–409, 2011) which proves the analogous result for the critical bond percolation for \(d\ge 11\), and by Werner (in: Séminaire de Probabilités XLVIII, Springer, Berlin, 2016) which conjectures the similarity between the GFF level-set and the bond percolation in general and proves this connection for various geometric aspects.