The coincidence of critical points in Poisson percolation models. (English. Russian original) Zbl 0661.60122

Men’shikov, M. V.; Sidorenko, A. F.

Theory Probab. Appl. 32, No. 3, 547-550 (1987); translation from Teor. Veroyatn. Primen. 32, No. 3, 603-606 (1987).

The authors consider a continuous percolation model in \({\mathbb{R}}^ d:\) percolation through clusters of “defects” of variable random shape of which the centers have Poisson distribution with intensity \(\lambda\). We can define two percolation tresholds \(\lambda_ T\) and \(\lambda_ H\). The first one is the point of divergence of mean cluster size and the second is the maximal value of \(\lambda\) for which the probability of existence of an unbounded cluster is equal to zero. For a wide class of situations the coincidence between \(\lambda_ T\) and \(\lambda_ H\) is proved (for example in the case of a finite number of shapes of defects).

Reviewer: P.A.Kučment

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B43	Percolation

Keywords:

percolation; percolation tresholds; existence of an unbounded cluster