A percolation process on the binary tree where large finite clusters are frozen

TL;DR

This paper investigates a percolation process on a binary tree where large clusters freeze, demonstrating convergence to a known process as the freezing threshold increases, and highlighting differences from similar processes on other lattices.

Contribution

It introduces a new percolation model on the binary tree with freezing, and proves its convergence to Aldous's frozen percolation process as the threshold grows.

Findings

Process converges to Aldous's frozen percolation as N increases

Behavior differs significantly from that on the square lattice

Large finite clusters freeze at a predictable rate

Abstract

We study a percolation process on the planted binary tree, where clusters freeze as soon as they become larger than some fixed parameter N. We show that as N goes to infinity, the process converges in some sense to the frozen percolation process introduced by Aldous. In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by van den Berg, de Lima and Nolin.