Abstract
The Marked Binary Branching Tree (MBBT) is the family tree of a rate one binary branching process, on which points have been generated according to a rate one Poisson point process, with i.i.d. uniformly distributed activation times assigned to the points. In frozen percolation on the MBBT, initially, all points are closed, but as time progresses points can become either frozen or open. Points become open at their activation times provided they have not become frozen before. Open points connect the parts of the tree below and above it and one says that a point percolates if the tree above it is infinite. We consider a version of frozen percolation on the MBBT in which at times of the form , all points that percolate are frozen. The limiting model for , in which points freeze as soon as they percolate, has been studied before by Ráth, Swart, and Terpai. We extend their results by showing that there exists a such that the model is endogenous for but not for . This means that for , frozen percolation is a.s. determined by the MBBT but for one needs additional randomness to describe it.
Funding Statement
The work of B. Ráth was partially supported by grant NKFI-FK-123962 of NKFI (National Research, Development and Innovation Office), the Bolyai Research Scholarship of the Hungarian Academy of Sciences, the ÚNKP-20-5-BME-5 New National Excellence Program of the Ministry for Innovation and Technology, and the ERC Synergy under Grant No. 810115 - DYNASNET. J.M. Swart is supported by grant 20-08468S of the Czech Science Foundation (GA CR). The work of Márton Szőke is partially supported by the ERC Consolidator Grant 772466 “NOISE”.
Acknowledgments
The authors would like to thank both anonymous referees for their numerous constructive comments that improved the quality of this paper.
Citation
Balázs Ráth. Jan M. Swart. Márton Szőke. "A phase transition between endogeny and nonendogeny." Electron. J. Probab. 27 1 - 43, 2022. https://doi.org/10.1214/22-EJP872
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