Arm Exponent for the Gaussian Free Field on Metric Graphs in Intermediate Dimensions

Abstract

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $\alpha$ and that the Green's function for the random walk on ${G}$ exhibits a power law decay with exponent $\nu$, in the regime $1\leq \nu \leq \frac{\alpha}{2}$. In particular, this includes the cases of ${G}=\mathbb Z^3$, for which $\nu=1$, and ${G}= \mathbb Z^4$, for which $\nu=\frac{\alpha}{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$ like $R^{-\frac{\nu}{2}+o(1)}$. Our results are in fact more precise and yield logarithmic corrections when $\nu > 1$ as well as corrections of order $\log \log R$ when $\nu=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu > 1$ and essentially optimal when $\nu=1$. This extends previous results from [16].