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This completes the proof of the proposition. 0 4 An Approximate Differential Equation for the Expected Number of Particles Per Site In this section we start with one particle at each site (~o = 11) and we write ~t instead of ~t(l1). We also do not write the superscript (=) to P and E in this section. We first derive a differential equation for E(t). J. van den Berg and H. 1. 1) Proof. This can be seen quite easily by a rather straightforward (first-order) bookkeeping of the particle movements (and their effects) to and from 0 in a small time interval.

14). 29) 26 J. van den Berg and H. Kesten of the terms with u x i= u. 30) u. 29) of the terms with u i= u. For the terms with u = one obtains similarly (again by replacing and then summing over x) the bound u C~~t L/1 2 (z) Lq(v) LP{v + S" and u' + S~" meet during p z xP{ v v + S~' and u' u by u' +x [O,A]} u' + S~" meet during [0, B I ]}. ,d, P{S~' and u+S~" meet during [O,A]} :S. P{ S: - S:' = u for some Cl4 < . 33) Randomly Coalescing Random Walk 27 Actually the estimate in Spitzer only holds for 3-dimensional random walk (see Uchiyama (1998)) and therefore should be applied to a triple of coordinates of the random walks {S"} and {S"'}.

1), of order exp( -clog A) since the boundary can be as short as O(log A). Once we establish that there is only a single droplet, it is of interest to study boundary fluctuations and boundary regularity for this droplet, as was done in [6] under different conditioning in infinite volume. This is mainly a matter of extending some of the results in [6] from infinite volume to finite volumes with wired boundary; this in turn involves showing that the boundary influence is negligible. 2 Definitions, Heuristics and Statement of Main Results The results in this paper make use of only a few basic properties of the FK or other percolation model, so we will state our results for general bond percolation models satisfying these properties.

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