Random spanning forests and hyperbolic symmetry

Abstract: The arboreal gas is the probability measure that arises from conditioning the random subgraph given by Bernoulli($p$) bond percolation to be a spanning forest, i.e., to contain no cycles. This conditioning makes sense on any finite graph $G$, and in the case $p=1/2$ gives the uniform measure on spanning forests. The arboreal gas also arises as a $q\to0$ limit of the $q$-state random cluster model.

What are the percolative properties of these forests? This turns out to be a surprisingly rich question, and I will discuss what is known and conjectured. I will also describe a 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 with Roland Bauerschmidt, Nick Crawford, and Andrew Swan.

dynamical systemsprobability

Audience: researchers in the topic

( slides | video )

Horowitz seminar on probability, ergodic theory and dynamical systems