Frozen percolation in two dimensions

TL;DR

This paper studies a modified percolation process on the triangular lattice where clusters freeze upon reaching a certain size, revealing that in the limit, clusters only freeze near the critical point, differing from Aldous' process.

Contribution

It introduces and analyzes a site version of frozen percolation on the triangular lattice, extending results to the square lattice and clarifying the process's behavior near criticality.

Findings

Clusters freeze only in the critical window as N approaches infinity.

The fraction of vertices in frozen clusters tends to zero as N increases.

The diameter of open clusters is typically smaller than N but of order N.

Abstract

Aldous introduced a modification of the bond percolation process on the binary tree where clusters stop growing (freeze) as soon as they become infinite. We investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach L infinity diameter at least N for some parameter N. We show that, informally speaking, in the limit N goes to infinity, the clusters only freeze in the critical window of site percolation on the triangular lattice. Hence the fraction of vertices that eventually (i. e. at time 1) are in a frozen cluster tends to 0 as N goes to infinity. We also show that the diameter of the open cluster at time 1 of a given vertex is, with high probability, smaller than N but of order N. This shows that the process on the triangular lattice has a behaviour quite different from Aldous' process. We also indicate which modifications have to be made to adapt the proofs to the case of the N-parameter frozen bond percolation process on the square lattice. This extends our results to the square lattice, and answers the questions posed by van den Berg, de Lima and Nolin in an earlier paper.