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

van den Berg, Jacob; Kiss, Demeter; Nolin, Pierre

Electron. Commun. Probab. 17, Paper No. 2, 11 p. (2012).

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