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Subcritical regimes in some models of continuum percolation. (English) Zbl 1172.60335
Summary: We consider some continuum percolation models. We are mainly interested in giving some sufficient conditions for the absence of percolation. We give some general conditions and then focus on two examples. The first one is a multiscale percolation model based on the Boolean model. It was introduced by R. Meester and R. Roy [Contiunuum percolation. Reprint of the 1996 hardback ed. Cambridge: Cambridge University Press (2008; Zbl 1146.60076)] and subsequently studied by M. V. Menshikov, S. Yu. Popov and M. Vachkovskaia [Probab. Theory Relat. Fields 119, No. 2, 176–186 (2001; Zbl 1001.60104), Bull. Braz. Math. Soc. (N.S.) 34, No. 3, 417–435 (2003; Zbl 1056.60099)]. The second one is based on the stable marriage of Poisson and Lebesgue introduced by C. Hoffman, A. E. Holroyd and Y. Peres [Ann. Probab. 34, No. 4, 1241–1272 (2006; Zbl 1111.60008)] and whose percolation properties have been studied by M. V. Freire, S. Popov and M. Vachkovskaia [Stochastic Processes Appl. 117, No. 4, 514–525 (2007; Zbl 1130.60092)].

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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