The Annals of Probability

Doob–Martin boundary of Rémy’s tree growth chain

Steven N. Evans, Rudolf Grübel, and Anton Wakolbinger

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Abstract

Rémy’s algorithm is a Markov chain that iteratively generates a sequence of random trees in such a way that the $n$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 $n$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. Two such ensembles represent the same point in the boundary if for each $m$ the random, rooted, planar, binary trees spanned by $m+1$ independent points chosen according to the respective probability measures have the same distribution. Also, the Doob–Martin boundary corresponds bijectively to the set of extreme point of the closed convex set of nonnegative harmonic functions that take the value $1$ at the binary tree with $3$ vertices; 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.

Article information

Source
Ann. Probab., Volume 45, Number 1 (2017), 225-277.

Dates
Received: November 2014
Revised: December 2015
First available in Project Euclid: 26 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1485421333

Digital Object Identifier
doi:10.1214/16-AOP1112

Mathematical Reviews number (MathSciNet)
MR3601650

Zentralblatt MATH identifier
06696270

Subjects
Primary: 60J50: Boundary theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 68W40: Analysis of algorithms [See also 68Q25]

Keywords
Binary tree tail $\sigma$-field Doob–Martin compactification Poisson boundary bridge real tree exchangeability continuum random tree Catalan number graph limit graphon partial order

Citation

Evans, Steven N.; Grübel, Rudolf; Wakolbinger, Anton. Doob–Martin boundary of Rémy’s tree growth chain. Ann. Probab. 45 (2017), no. 1, 225--277. doi:10.1214/16-AOP1112. https://projecteuclid.org/euclid.aop/1485421333


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