A Markov chain on binary trees
Consider a Markov process in the space of binary trees - a randomly changing tree - in which, at each step, you delete a random leaf and then grow a new leaf in a random location on the tree. This chain is used in a Markov chain Monte Carlo algorithm for phylogenetic inference. We will discuss a method of swapping leaf labels to obtain a family of projectively consistent Markov chains on binary trees with edge weights, with the aim of understanding the (limiting) behavior of these chains on very large binary trees. This label swapping has been used to resolve a 2000 conjecture of Aldous on the scaling limit of the Markov chain. This is joint work with Soumik Pal, Douglas Rizzolo, and Matthias Winkel.