2018/02/02 by Forman, Noah, Pal, Soumik, Rizzolo, Douglas +1
#60C05 #60J10 #60J80 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1802.00862
Consider the Aldous Markov chain on the space of rooted binary trees with n labeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix 1≤ k < n and project the leaf mass onto the subtree spanned by the first k leaves. This yields a binary tree with edge weights that we call a "decorated k-tree with total mass n." We introduce label swapping dynamics for the Aldous chain so that, when it runs in stationarity, the decorated k-trees evolve as Markov chains themselves, and are projectively consistent over k≤ n. The construction of projectively consistent chains is a crucial step in the construction of the Aldous diffusion on continuum trees by the present authors, which is the n→ ∞ continuum analogue of the Aldous chain and will be taken up elsewhere. Some of our results have been generalized to Ford's alpha model trees.