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

For massless vertex-transitive transient graphs, the percolation phase transition for level sets of Gaussian free field on associated continuous cable system is particularly well understood, and in particular critical parameter h˜∗ always equal to zero. On general two weak conditions graph G are given [12], each which implies one inequalities h˜∗≤0 h˜∗≥0. In this article, we give counterexamples show that none these necessary, prove strict inequality h˜∗<0 typical massive graphs with bounded weights, provide an example a h˜∗=∞. way, obtain another characterization random interlacements as isomorphism between Doob h-transform interlacements, two-dimensional pinned interlacements.