Projections of the Aldous chain on binary trees: Intertwining and consistency

Noah Mills Forman, Soumik Pal, Douglas Rizzolo, Matthias Winkel

Abstract:

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 dec-
orated k-trees evolve as Markov chains themselves, and are
projectively consistent over k. 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 analog of the Aldous chain
and will be taken up elsewhere.