The Aldous Chain On Cladograms In The Diffusion Limit

In (Combin. Probab. Comput. 9 (2000) 191-204), Aldous investigates a symmetric Markov chain on cladograms and gives bounds on its mixing and relaxation times. The latter bound was sharpened in (Random Structures Algorithms 20 (2002) 59-70). In the present paper, we encode cladograms as binary, algebraic measure trees and show that this Markov chain on cladograms with a fixed number of leaves converges in distribution as the number of leaves tends to infinity. We give a rigorous construction of the limit as the solution of a well-posed martingale problem. The existence of a continuum limit diffusion was conjectured by Aldous, and we therefore refer to it as Aldous diffusion. We show that the Aldous diffusion is a Feller process with continuous paths, and the algebraic measure Brownian CRT is its unique invariant distribution.Furthermore, we consider the vector of the masses of the three subtrees connected to a sampled branch point. In the Brownian CRT, its annealed law is known to be the Dirichlet distribution. Here, we give an explicit expression for the infinitesimal evolution of its quenched law under the Aldous diffusion.