Percolation for the Gaussian free field on the cable system: counterexamples

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.