Doob--Martin boundary of R\'emy's tree growth chain

TL;DR

This paper characterizes the Doob--Martin boundary of Rémys' tree growth chain, describing how the process can be conditioned to 'go to infinity' using a novel boundary representation involving real trees and probability measures.

Contribution

It provides a concrete description of the Doob--Martin boundary for Rémys' tree growth chain, linking it to real trees, measures, and kernels, extending the theory of graph limits.

Findings

Boundary points correspond to real trees with measures and kernels.

Convergence of finite trees to boundary points is characterized by distributional limits.

The boundary is in bijection with extreme points of harmonic functions.

Abstract

R\'emy's algorithm is a Markov chain that iteratively generates a sequence of random trees in such a way that the $n^{\mathrm{th}}$ tree is uniformly distributed over the set of rooted, planar, binary trees with $2n+1$ vertices. We obtain a concrete characterization of the Doob--Martin boundary of this transient Markov chain and thereby delineate all the ways in which, loosely speaking, this process can be conditioned to "go to infinity" at large times. A (deterministic) sequence of finite rooted, planar, binary trees converges to a point in the boundary if for each $m$ the random rooted, planar, binary tree spanned by $m+1$ leaves chosen uniformly at random from the $n^{\mathrm{th}}$ tree in the sequence converges in distribution as $n$ tends to infinity -- a notion of convergence that is analogous to one that appears in the recently developed theory of graph limits. We show that a point in the Doob--Martin boundary may be identified with the following ensemble of objects: a complete separable $\mathbb{R}$-tree that is rooted and binary in a suitable sense, a diffuse probability measure on the $\mathbb{R}$-tree that allows us to make sense of sampling points from it, and a kernel on the $\mathbb{R}$-tree that describes the probability that the first of a given pair of points is below and to the left of their most recent common ancestor while the second is below and to the right. The Doob--Martin boundary corresponds bijectively to the set of extreme points of the closed convex set of normalized nonnegative harmonic functions, in other words, the minimal and full Doob--Martin boundaries coincide. These results are in the spirit of the identification of graphons as limit objects in the theory of graph limits.