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

TL;DR

This paper studies a Markov chain on binary trees, introduces a projection method to create consistent subtrees, and develops a framework that supports the continuum limit leading to Aldous diffusion.

Contribution

It introduces label swapping dynamics for the Aldous chain, ensuring projective consistency of subtree projections, facilitating the construction of the Aldous diffusion on continuum trees.

Findings

Decorated $k$-trees evolve as Markov chains in stationarity.

The chain construction is projectively consistent over different $k$.

Results extend to Ford's alpha model trees.

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 $1\le k < n$ 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.