Frozen percolation on the binary tree is nonendogenous

TL;DR

This paper proves that the frozen percolation process on the binary tree is nonendogenous, meaning the freezing event cannot be determined solely by the assigned activation times, using a scale-invariant Galton-Watson tree analysis.

Contribution

It demonstrates that the recursive tree process for frozen percolation on the binary tree is nonendogenous, answering a question posed by Aldous and Bandyopadhyay.

Findings

Frozen percolation on the binary tree is nonendogenous.

A scale-invariant Galton-Watson tree model is used in the proof.

The process's freezing events are not measurable functions of activation times.

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 endvertices is part of an infinite open cluster; in the opposite case, it freezes. Aldous (2000) 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 (2005), 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.