Reversing the cut tree of the Brownian continuum random tree

Abstract

Consider the Aldous–Pitman fragmentation process [7] of a Brownian continuum random tree $\mathcal{T} ^{\mathrm{br} }$. The associated cut tree $\operatorname{cut} (\mathcal{T} ^{\mathrm{br} })$, introduced by Bertoin and Miermont [13], 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 $\operatorname{cut} (\mathcal{T} ^{\mathrm{br} })$.