A growth-fragmentation-isolation process on random recursive trees

Bansaye, Vincent; Gu, Chenlin; Yuan, Linglong

We consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters are frozen with a rate proportional to their size (isolation of connected component). A phase transition occurs when the isolation is able to stop the growth fragmentation process and cause extinction. When the process survives, we characterize its growth and prove that the empirical measure of clusters a.s. converges to a limit law on recursive trees. This approach exploits the branching structure associated to the size of clusters, which is inherited from the splitting property of random recursive trees. This issue is motivated by the control of epidemics and contact-tracing where clusters correspond to subtrees of infected individuals that can be identified and isolated.