# 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 $\Lambda$-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 θ ≤ θ ∈ [0, ∞], so that if θ^{\star} < 1, the process comes down from infinity and if θ_{\star} > 1, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters θ^{\star} , θ_{\star} coincide and are explicit.
Keywords :
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03211894
Contributor : Clément Foucart <>
Submitted on : Thursday, April 29, 2021 - 10:20:20 AM
Last modification on : Wednesday, June 2, 2021 - 3:28:23 AM

### File

AAP1691.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-03211894, version 1

### Citation

Clément Foucart. A PHASE TRANSITION IN THE COMING DOWN FROM INFINITY OF SIMPLE EXCHANGEABLE FRAGMENTATION-COAGULATION PROCESSES. 2021. ⟨hal-03211894⟩

Record views