  For massless vertex-transitive transient graphs, the percolation phase
transition for the level sets of the Gaussian free field on the associated
continuous cable system is particularly well understood, and in particular the
associated critical parameter $\widetilde{h}_*$ is always equal to zero. On
general transient graphs, two weak conditions on the graph $\mathcal{G}$ are
given in arXiv:2101.05800, each of which implies one of the two inequalities
$\widetilde{h}_*\leq0$ and $\widetilde{h}_*\geq0.$ In this article, we give two
counterexamples to show that none of these two conditions are necessary, prove
that the strict inequality $\widetilde{h}_*<0$ is typical on massive graphs
with bounded weights, and provide an example of a graph on which
$\widetilde{h}_*=\infty.$ On the way, we obtain another characterization of
random interlacements on massive graphs, as well as an isomorphism between the
Gaussian free field and the Doob $\mathit{\mathbf{h}}$-transform of random
interlacements, and between the two-dimensional pinned free field and random
interlacements.
