A percolation process on the square lattice where large finite clusters are frozen

In (Aldous, Math. Proc. Cambridge Philos. Soc. 128 (2000), 465-477), Aldous constructed a growth process for the binary tree where clusters freeze as soon as they become infinite. It was pointed out by Benjamini and Schramm that such a process does not exist for the square lattice. This motivated us to investigate the modified process on the square lattice, where clusters freeze as soon as they have diameter larger than or equal to N, the parameter of the model. The non-existence result, mentioned above, raises the question if the N- parameter model shows some 'anomalous' behaviour as N →∞. For instance, if one looks at the cluster of a given vertex, does, as N →∞, the probability that it eventually freezes go to 1? Does this probability go to 0? More generally, what can be said about the size of a final cluster? We give a partial answer to some of such questions.