  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. Under the sole assumption that its sign clusters do not
percolate, we derive an extension of Lupu's formula for the two-point function
at criticality. We then focus on the low-dimensional case $0< \nu <
\frac{\alpha}{2}$, where $\alpha$ governs the polynomial volume growth of $G$
and $\nu$ the decay rate of the Green's function on $G$. In particular, this
includes the benchmark case ${G}=\mathbb{Z}^3$, for which $\alpha=3$ and $\nu=
\alpha-2=1$. We prove under these assumptions that the critical one-arm
probability decays with distance $R$ like $R^{-\frac{\nu}{2}}$, up to
multiplicative constants.
