Frozen percolation on the triangular lattice

Aldous introduced a growth model for the binary tree where clusters freeze as soon as they become infinite. Benjamini and Schramm showed that such process does not exist on the triangular lattice. To circumvent the problem of non-existence, van den Berg, de Lima and Nolin modified the process so that the clusters freeze as soon as their diameter reach N, the parameter of the model. They raised the question whether the probability that a given vertex eventually freezes in the N-parameter model tends to 0 as N goes to infinity. We answer this question in the affirmative by showing that for all k>0, in the N-parameter frozen percolation process the probability that all the frozen clusters in a box of size kN form close to time 1/2 tends to 1 as N goes to infinity. We also highlight some connections between frozen percolation processes and forest fire models.