Abstract: The arboreal gas is the probability measure that arises from conditioning the random subgraph given by Bernoulli(
) bond percolation to be a spanning forest, i.e., to contain no cycles. This conditioning makes sense on any finite graph 
, and in the case 
gives the uniform measure on spanning forests. The arboreal gas also arises as a 
limit of the 
-state random cluster model.
What are the percolative properties of these forests? This turns out to be a surprisingly rich question. The first talk (Helmuth) will discuss what is known rigorously and the conjectural behaviour on 
. The second talk (Bauerschmidt) will introduce an important tool for studying connection probabilities, the magic formula, which arises due to an important connection between the arboreal gas and spin systems with hyperbolic symmetry.
Based on joint work of Roland Bauerschmidt, Nick Crawford, Tyler Helmuth, and Andrew Swan.