  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.
