Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Inverting the cut-tree transform

Louigi Addario-Berry, Daphné Dieuleveut, and Christina Goldschmidt

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Abstract

We consider fragmentations of an $\mathbb{R}$-tree $T$ driven by cuts arriving according to a Poisson process on $T\times[0,\infty)$, where the first co-ordinate specifies the location of the cut and the second the time at which it occurs. The genealogy of such a fragmentation is encoded by the so-called cut-tree, which was introduced by Bertoin and Miermont (Ann. Appl. Probab. 23 (4) (2013) 1469–1493) for a fragmentation of the Brownian continuum random tree. The cut-tree was generalised by Dieuleveut (Ann. Appl. Probab. 25 (4) (2015) 2215–2262) to a fragmentation of the $\alpha$-stable trees, $\alpha\in(1,2)$, and by Broutin and Wang (Bernoulli 23 (4A) (2017) 2380–2433) to the inhomogeneous continuum random trees of Aldous and Pitman (Probab. Theory Related Fields 118 (4) (2000) 455–482). In the first two cases, the projections of the forest-valued fragmentation processes onto the sequence of masses of their constituent subtrees yield an important family of examples of Bertoin’s self-similar fragmentations (Ann. Inst. Henri Poincaré Probab. Stat. 38 (3) (2002) 319–340); in the first case the time-reversal of the fragmentation gives an additive coalescent. Remarkably, in all of these cases, the law of the cut-tree is the same as that of the original $\mathbb{R}$-tree.

In this paper, we develop a clean general framework for the study of cut-trees of $\mathbb{R}$-trees. We then focus particularly on the problem of reconstruction: how to recover the original $\mathbb{R}$-tree from its cut-tree. This has been studied in the setting of the Brownian CRT by Broutin and Wang (Electron. J. Probab. 22 (2017) 80), who prove that it is possible to reconstruct the original tree in distribution. We describe an enrichment of the cut-tree transformation, which endows the cut-tree with information we call a consistent collection of routings. We show this procedure is well-defined under minimal conditions on the $\mathbb{R}$-trees. We then show that, for the case of the Brownian CRT and the $\alpha$-stable trees with $\alpha\in(1,2)$, the original tree and the Poisson process of cuts thereon can both be almost surely reconstructed from the enriched cut-trees. For the latter results, our methods make essential use of the self-similarity and re-rooting invariance of these trees.

Résumé

Nous considérons des fragmentations d’un $\mathbb{R}$-arbre $T$ dirigées par des coupures qui arrivent selon un processus de Poisson sur $T\times[0,\infty)$, où la première composante désigne le point auquel se produit la coupure et le deuxième, l’instant auquel elle a lieu. La généalogie d’une telle fragmentation est codée par l’arbre des coupes, qui a été introduit par Bertoin et Miermont (Ann. Appl. Probab. 23 (4) (2013) 1469–1493) pour une fragmentation de l’arbre brownien. L’arbre des coupes a ensuite été généralisé par Dieuleveut (Ann. Appl. Probab. 25 (4) (2015) 2215–2262) à une fragmentation des arbres $\alpha$-stables, pour $\alpha\in(1,2)$, et par Broutin et Wang (Bernoulli 23 (4A) (2017) 2380–2433) aux arbres continus inhomogènes d’Aldous et Pitman (Probab. Theory Related Fields 118 (4) (2000) 455–482). Dans les deux premiers cas, l’évolution de la suite des masses des sous-arbres apparaissant dans le processus de fragmentation constitue une famille importante de processus de fragmentation auto-similaires de Bertoin (Ann. Inst. Henri Poincaré Probab. Stat. 38 (3) (2002) 319–340); dans le premier cas, une fois inversée dans le temps, la fragmentation devient un coalescent additif. Remarquablement, dans tous ces cas, la loi de l’arbre des coupes est la même que celle du $\mathbb{R}$-arbre initial.

Dans cet article, nous développons un cadre général pour l’étude de l’arbre des coupes d’un $\mathbb{R}$-arbre. Par la suite, nous nous concentrons particulièrement sur le problème de la reconstruction : comment retrouver le $\mathbb{R}$-arbre initial à partir de son arbre des coupes. Ce problème a été étudié pour l’arbre brownien par Broutin et Wang (Electron. J. Probab. 22 (2017) 80), qui démontrent qu’il est possible de reconstruire l’arbre initial en loi. Nous décrivons un enrichissement de la construction de l’arbre des coupes, qui dote l’arbre des coupes d’une structure supplémentaire que nous appelons une collection cohérente de routages. Nous démontrons que ce procédé est bien défini sous des conditions minimales sur le $\mathbb{R}$-arbre. Ensuite, nous démontrons que pour l’arbre brownien ainsi que pour l’arbre $\alpha$-stable avec $\alpha\in(1,2)$, l’arbre initial muni de son processus de Poisson de coupures peut être reconstruit presque sûrement à partir de l’arbre des coupes enrichi. Pour ces derniers résultats, nous utilisons de façon essentielle l’auto-similarité et l’invariance par réenracinement de ces $\mathbb{R}$-arbres aléatoires.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1349-1376.

Dates
Received: 11 July 2016
Revised: 14 March 2018
Accepted: 18 July 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398872

Digital Object Identifier
doi:10.1214/18-AIHP921

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 05C05: Trees 60G52: Stable processes 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
$\mathbb{R}$-tree Cut-tree Fragmentation

Citation

Addario-Berry, Louigi; Dieuleveut, Daphné; Goldschmidt, Christina. Inverting the cut-tree transform. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1349--1376. doi:10.1214/18-AIHP921. https://projecteuclid.org/euclid.aihp/1569398872


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