The Arboreal Gas
In Bernoulli bond percolation each edge of a graph is declared open with probability p (and closed otherwise) and one typically asks questions about the random subgraph of open edges. The arboreal gas is the probability measure obtained by conditioning on the event that the subgraph of open edges is a forest, i.e., contains no cycles.
What are the percolative properties of these random forests? Do they contain giant trees? This turns out to be a surprisingly rich question. I will discuss what is known and some conjectures.Â