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.
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.
We thank James Martin for useful discussions and Márton Szőke for help with the simulations.
"Frozen percolation on the binary tree is nonendogenous." Ann. Probab. 49 (5) 2272 - 2316, September 2021. https://doi.org/10.1214/21-AOP1507