September 2021 Frozen percolation on the binary tree is nonendogenous
Balázs Ráth, Jan M. Swart, Tamás Terpai
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Ann. Probab. 49(5): 2272-2316 (September 2021). DOI: 10.1214/21-AOP1507

Abstract

In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time an edge opens provided neither of its end vertices is part of an infinite open cluster; in the opposite case it freezes. Aldous (Math. Proc. Cambridge Philos. Soc. 128 (2000) 465–477) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay (Ann. Appl. Probab. 15 (2005) 1047–1110), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton–Watson tree that has nice scale invariant properties.

Funding Statement

The first author was partially supported by Postdoctoral Fellowship NKFI-PD-121165 and Grant NKFI-FK-123962 of NKFI (National Research, Development and Innovation Office), the Bolyai Research Scholarship of the Hungarian Academy of Sciences and the ÚNKP-19-4-BME-85 New National Excellence Program of the Ministry for Innovation and Technology.
The second author was supported by Grant 19-07140S of the Czech Science Foundation (GA CR).
The third author was partially supported by the National Research, Development and Innovation Office NKFIH Grant K 120697.

Acknowledgements

We thank James Martin for useful discussions and Márton Szőke for help with the simulations.

Citation

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Balázs Ráth. Jan M. Swart. Tamás Terpai. "Frozen percolation on the binary tree is nonendogenous." Ann. Probab. 49 (5) 2272 - 2316, September 2021. https://doi.org/10.1214/21-AOP1507

Information

Received: 1 November 2019; Revised: 1 December 2020; Published: September 2021
First available in Project Euclid: 24 September 2021

Digital Object Identifier: 10.1214/21-AOP1507

Subjects:
Primary: 82C27
Secondary: 60J80 , 60K35 , 82C26

Keywords: branching process , endogeny , frozen percolation , Near-critical percolation , recursive distributional equation , recursive tree process , self-organised criticality

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • September 2021
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