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 α and that the Green's function for the random walk on G exhibits a power law decay with exponent ν, in the regime 1≤ν≤α/2. In particular, this includes the cases of G=Z3, for which ν=1, and G=Z4, for which ν=α2=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−ν2+o(1). Our results are in fact more precise and yield logarithmic corrections when ν>1 as well as corrections of order loglogR when ν=1. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when ν>1 and essentially optimal when ν=1. This extends previous results from [16].