The wired arboreal gas on regular trees

TL;DR

This paper investigates the limiting behavior of the arboreal gas on regular trees, revealing a phase transition and its relation to bond percolation, with a unique supercritical phase exhibiting critical-like properties.

Contribution

It establishes the existence and independence of the weak limit of the arboreal gas on regular trees and characterizes its phases, including a novel supercritical phase with critical-like behavior.

Findings

Limit exists and is independent of exhaustion

Model is equivalent to bond percolation at and below criticality

Supercritical phase combines bond percolation with infinite paths

Abstract

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.