We study the weak limit of the arboreal gas along any exhaustion of a regular tree with wired boundary conditions. We prove that this limit exists, does not depend on the choice of exhaustion, and undergoes a phase transition. Below and at criticality, we prove the model is equivalent to bond percolation. Above criticality, we characterise the model as the superposition of critical bond percolation and a random collection of infinite one-ended paths. This provides a simple example of an arboreal gas model that continues to exhibit critical-like behaviour throughout its supercritical phase.
We thank Tom Hutchcroft for suggesting this model and for many valuable comments. We also thank Roland Bauerschmidt for helpful feedback on our presentation of an earlier version of these results. We were supported by an EPSRC DTP grant.
"The wired arboreal gas on regular trees." Electron. Commun. Probab. 27 1 - 10, 2022. https://doi.org/10.1214/22-ECP460