Frozen percolation in two dimensions

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.