  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 tends to infinity. For
instance, if one looks at the cluster of a given vertex, does, as N tends to
infinity, 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.
