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

TL;DR

This paper investigates the conditions under which exchangeable fragmentation-coagulation processes starting with infinitely many blocks can reduce to finitely many blocks, identifying a phase transition governed by specific parameters.

Contribution

It introduces sharp parameters that determine whether such processes come down from infinity or stay infinite, providing explicit results for regularly varying measures.

Findings

Processes come down from infinity if parameter ^{\u2212}<1

Processes stay infinite if parameter ^{}>1

Explicit parameters for regularly varying measures

Abstract

We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $\Lambda$-coalescent, and fragmentation dislocates at 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 $\theta_{\star}\leq \theta^{\star}\in [0,\infty]$, so that if $\theta^{\star}<1$, the process comes down from infinity and if $\theta_\star>1$, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters $\theta_{\star},\theta^{\star}$ coincide and are explicit.