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Inverting the cut-tree transform. (English. French summary) Zbl 1467.60014
Summary: 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 J. Bertoin and G. Miermont [Ann. Appl. Probab. 23, No. 4, 1469–1493 (2013; Zbl 1279.60035)] for a fragmentation of the Brownian continuum random tree. The cut-tree was generalised by D. Dieuleveut [Ann. Appl. Probab. 25, No. 4, 2215–2262 (2015; Zbl 1319.60167)] to a fragmentation of the \(\alpha \)-stable trees, \( \alpha\in(1,2)\), and by N. Broutin and M. Wang [Bernoulli 23, No. 4A, 2380–2433 (2017; Zbl 1388.60036)] to the inhomogeneous continuum random trees of D. Aldous and J. Pitman [Probab. Theory Relat. Fields 118, No. 4, 455–482 (2000; Zbl 0969.60015)]. 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 [J. Bertoin, Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 3, 319–340 (2002; Zbl 1002.60072)]; 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 N. Broutin and M. Wang [Electron. J. Probab. 22, Paper No. 80, 23 p. (2017; Zbl 1379.60095)], 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.
MSC:
60C05 Combinatorial probability
05C05 Trees
60G52 Stable stochastic processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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