Reversing the cut tree of the Brownian continuum random tree

Abstract : Consider the logging process of the Brownian continuum random tree (CRT) $\cal T$ using a Poisson point process of cuts on its skeleton [Aldous and Pitman, Ann. Probab., vol. 26, pp. 1703--1726, 1998]. Then, the cut tree introduced by Bertoin and Miermont describes the genealogy of the fragmentation of $\cal T$ into connected components [Ann. Appl. Probab., vol. 23, pp. 1469--1493, 2013]. This cut tree cut$(\cal T)$ is distributed as another Brownian CRT, and is a function of the original tree $\cal T$ and of the randomness in the logging process. We are interested in reversing the transformation of $\cal T$ into cut$(\cal T)$: we define a shuffling operation, which given a Brownian CRT $\cal H$, yields another one shuff$(\cal H)$ distributed in such a way that $(\cal T$,cut$(\cal T))$ and $($shuff$(\cal H), \cal H)$ have the same distribution.
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Submitted on : Friday, August 15, 2014 - 1:04:04 PM
Last modification on : Thursday, February 7, 2019 - 5:49:32 PM

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  • HAL Id : hal-01056124, version 1
  • ARXIV : 1408.2924

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Nicolas Broutin, Minmin Wang. Reversing the cut tree of the Brownian continuum random tree. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2017, 22 (80), pp.1-23. ⟨hal-01056124⟩

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