The arboreal gas, alternatively known as the edge weighted unrooted spanning forest model, is equivalent to Bernoulli percolation conditioned to be acyclic. Recent exciting work of Bauerschmidt, Crawford and Helmuth has established in dimensions d>2 the existence of a value for the edge weight parameter above which certain infinite volume limits contain an infinite tree almost surely, and thus that the model demonstrates a phase transition with respect to this parameter. Using probabilistic techniques, we show that in low dimensions d=3,4 the infinite tree is unique, and give strong heuristic evidence that the number of infinite trees is in fact infinite in higher dimensions. We also prove that in any dimension all such infinite trees must be one-ended almost surely. Joint work with Tom Hutchcroft.
The talk is based on https://arxiv.org/abs/2302.12224