A phase transition in the coming down from infinity of simple exchangeable fragmentation-coagulation processes

Abstract

We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a Λ-coalescent, and fragmentation dislocates at a finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters ${\mathit{\theta }}_{\star }\le {\mathit{\theta }}^{\star }\in [0,\infty ]$, so that if ${\mathit{\theta }}^{\star }<1$, the process comes down from infinity and if ${\mathit{\theta }}_{\star }>1$, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters ${\mathit{\theta }}^{\star }$, ${\mathit{\theta }}_{\star }$ coincide and are explicit.