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

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 $1\le k < n$ 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 $k\le n$. 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 $n\rightarrow \infty$ 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.