PROBLEM. Give conditions on the rate function r which are sufficient for percolation, and give conditions on the rate function r which are sufficient for non-percolation.
PROBLEM. In particular, for r = 1 does percolation occur?
DISCUSSION. On one hand, the particular case
(1) r(A,B) = length of boundary between A and B
is essentially just the usual bond percolation process on the
planar dual graph, and so percolation does occur.
On the other hand, consider a statistic such as
History. Problem discussed with Vlada Limic in 2001. In the case r = 1 we conjecture that percolation does occur. This case has the special property that each piece of the original (t = 0) boundary is present at time t with probability $e^{-t}$. The standard Peierls contour method is almost enough to prove this conjecture, but we could never quite finish the details of the estimates.
Update (10/2009). The preceding material has been written up as this short paper but the problems remain open.
Update (6/2017). A related process is studied in Random partitions of the plane via Poissonian coloring, and a self-similar process of coalescing planar partitions.