## A percolation process on the binary tree where large finite clusters are frozen.(English)Zbl 1244.60098

Summary: We study a percolation process on the planted binary tree, where clusters freeze as soon as they become larger than some fixed parameter $$N$$. We show that as $$N$$ goes to infinity, the process converges in some sense to the frozen percolation process introduced by David J. Aldous [Math. Proc. Camb. Philos. Soc. 128, No. 3, 465–477 (2000; Zbl 0961.60096)]. In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by J. van den Berg, B. N. B. de Lima and P. Nolin [Random Struct. Algorithms 40, No. 2, 220–226 (2012; Zbl 1235.60144)].

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation 05C80 Random graphs (graph-theoretic aspects)

### Keywords:

percolation; frozen cluster

### Citations:

Zbl 0961.60096; Zbl 1235.60144
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