Reversing the cut tree of the Brownian continuum random tree

International audienceConsider 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