## The Annals of Probability

### Doob–Martin boundary of Rémy’s tree growth chain

#### 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
Revised: December 2015
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.aop/1485421333

Digital Object Identifier
doi:10.1214/16-AOP1112

Mathematical Reviews number (MathSciNet)
MR3601650

Zentralblatt MATH identifier
06696270

#### 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

#### References

• [1] Aldous, D. (1991). The continuum random tree. I. Ann. Probab. 19 1–28.
• [2] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
• [3] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
• [4] Alonso, L. and Schott, R. (1995). Random Generation of Trees. Random Generators in Computer Science. Kluwer Academic, Boston, MA.
• [5] Austin, T. (2008). On exchangeable random variables and the statistics of large graphs and hypergraphs. Probab. Surv. 5 80–145.
• [6] Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353–355.
• [7] Borgs, C., Chayes, J. and Lovász, L. (2010). Moments of two-variable functions and the uniqueness of graph limits. Geom. Funct. Anal. 19 1597–1619.
• [8] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
• [9] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
• [10] Chen, B. and Winkel, M. (2013). Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees. Ann. Inst. Henri Poincaré Probab. Stat. 49 839–872.
• [11] Diaconis, P. and Janson, S. (2008). Graph limits and exchangeable random graphs. Rend. Mat. Appl. (7) 28 33–61.
• [12] Doob, J. L. (1959). Discrete potential theory and boundaries. J. Math. Mech. 8 433–458; erratum 993.
• [13] Evans, S. N. (2008). Probability and Real Trees. Lecture Notes in Math. 1920. Springer, Berlin.
• [14] Evans, S. N., Grübel, R. and Wakolbinger, A. (2012). Trickle-down processes and their boundaries. Electron. J. Probab. 17 no. 1, 58.
• [15] Föllmer, H. (1975). Phase transition and Martin boundary. In Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, Années Universitaires 1973/1974 et 1974/1975). Lecture Notes in Math. 465 305–317. Springer, Berlin.
• [16] Grübel, R. (2015). Persisting randomness in randomly growing discrete structures: Graphs and search trees. Discrete Math. Theor. Comput. Sci. 18 1–23.
• [17] Gufler, S. (2016). A representation for exchangeable coalescent trees and generalized tree-valued Fleming–Viot processes. Preprint. Available at arXiv:1608.08074.
• [18] Janson, S. (2011). Poset limits and exchangeable random posets. Combinatorica 31 529–563.
• [19] Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York.
• [20] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1976). Denumerable Markov Chains, 2nd ed. Springer, New York.
• [21] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Birkhäuser, Basel.
• [22] Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society Colloquium Publications 60. Amer. Math. Soc., Providence, RI.
• [23] Lovász, L. and Szegedy, B. (2006). Limits of dense graph sequences. J. Combin. Theory Ser. B 96 933–957.
• [24] Luczak, M. and Winkler, P. (2004). Building uniformly random subtrees. Random Structures Algorithms 24 420–443.
• [25] Marchal, P. (2003). Constructing a sequence of random walks strongly converging to Brownian motion. In Discrete Random Walks (Paris, 2003). Discrete Math. Theor. Comput. Sci. Proc., AC 181–190 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
• [26] Noah, C., Haulk, C. and Pitman, J. (2011). A representation of exchangeable hierarchies by sampling from real trees. Preprint. Available at arXiv:1101.5619.
• [27] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Springer, Berlin.
• [28] Rémy, J.-L. (1985). Un procédé itératif de dénombrement d’arbres binaires et son application à leur génération aléatoire. RAIRO. Inform. Théor. 19 179–195.
• [29] Revuz, D. (1975). Markov Chains. North-Holland Mathematical Library 11. North-Holland, Amsterdam.
• [30] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Cambridge Mathematical Library 1. Cambridge Univ. Press, Cambridge. Foundations, Reprint of the second (1994) edition.
• [31] Sawyer, S. A. (1997). Martin boundaries and random walks. In Harmonic Functions on Trees and Buildings (New York, 1995). Contemp. Math. 206 17–44. Amer. Math. Soc., Providence, RI.
• [32] Stanley, R. P. (1997). Enumerative Combinatorics. Vol. 1. Cambridge Studies in Advanced Mathematics 49. Cambridge Univ. Press, Cambridge.
• [33] Tao, T. (2007). A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma. J. Anal. Math. 103 1–45.
• [34] von Weizsäcker, H. (1983). Exchanging the order of taking suprema and countable intersections of $\sigma$-algebras. Ann. Inst. H. Poincaré Sect. B (N.S.) 19 91–100.
• [35] Woess, W. (2009). Denumerable Markov Chains. Generating Functions, Boundary Theory, Random Walks on Trees. European Mathematical Society (EMS), Zürich.