Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions

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 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 arXiv:2101.05801 and arXiv:1807.11117.