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Reversing the cut tree of the Brownian continuum random tree. (English) Zbl 1379.60095
Summary: Consider the Aldous-Pitman fragmentation process [D. Aldous and J. Pitman, Ann. Probab. 26, No. 4, 1703–1726 (1998; Zbl 0936.60064)] of a Brownian continuum random tree \(\mathcal{T}^{\mathrm{br}}\). The associated cut tree \(\operatorname{cut}(\mathcal{T}^{\mathrm{br}})\), introduced by J. Bertoin and G. Miermont [ibid. 23, No. 4, 1469–1493 (2013; Zbl 1279.60035)], is defined in a measurable way from the fragmentation process, describing the genealogy of the fragmentation, and is itself distributed as a Brownian CRT. In this work, we introduce a shuffle transform, which can be considered as the reverse of the map taking \(\mathcal{T}^{\mathrm{br}}\) to \(\operatorname{cut}(\mathcal{T}^{\mathrm{br}})\).

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60C05 Combinatorial probability
60G18 Self-similar stochastic processes
60F15 Strong limit theorems
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