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

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.