The wired arboreal gas on regular trees

We study the weak limit of arboreal gas along any exhaustion a regular tree with wired boundary conditions. prove that this exists, does not depend on choice exhaustion, and undergoes phase transition. Below at criticality, we model is equivalent to bond percolation. Above characterise as superposition critical percolation random collection infinite one-ended paths. This provides simple example an continues exhibit critical-like behaviour throughout its supercritical phase.