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The percolation process on a tree where inifinite clusters are frozen. (English) Zbl 0961.60096
Author’s 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 \(t> {1\over 2}\) 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 \(t>{1\over 2}\) has the distribution of a critical percolation cluster. This elaborates an observation of W. H. Stockmayer [J. Chem. Phys. 11, 45-55 (1943)].

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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