Title: The Percolation Process on a Tree Where Infinite Clusters are Frozen.
Author: David J. Aldous
Abstract:
Modify the usual percolation process on the infinite binary tree
by forbidding infinite clusters to grow further.
The ultimate configuration will consist of both infinite and finite
clusters. We give a rigorous construction of a version of this
process and show that one can do explicit calculations
of various quantities, for instance the law of the time (if any)
that the cluster containing a fixed edge becomes infinite.
Surprisingly, the distribution of the shape of a cluster which
becomes infinite at time
$t>1/2$ does not depend on $t$; it is always distributed as the
incipient infinite percolation cluster on the tree.
Similarly, a typical finite cluster at each time $t>1/2$ has the
distribution of a critical percolation cluster.
This elaborates an observation of Stockmayer (1942).