The Russo-Seymour-Welsh theorem and the equality of critical densities and the “dual” critical densities for continuum percolation on \({\mathbb{R}}^ 2\). (English) Zbl 0719.60119

Roy, Rahul

Ann. Probab. 18, No. 4, 1563-1575 (1990).

The paper studies a continuum percolation process in which discs of random (but bounded) radius are centered at the points of a Poisson process with intensity \(\lambda\). There are various metric characteristics associated with the cluster of discs containing the origin, such as its volume or diameter. Using results of M. V. Men’shikov, S. A. Molchanov and A. F. Sidorenko [Itogi Nauki Tekh., Ser. Teor. Veroyatn., Mat. Stat., Teor. Kibern. 24, 53-110 (1986; Zbl 0647.60103)], the author shows that all such quantities exhibit a phase transition at the same critical point \(\lambda_ c\). Furthermore, an RSW lemma is established in two dimensions, dealing with concatenations of vacant crossings of rectangles.

Reviewer: G.Grimmett (Bristol)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B43	Percolation
82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K10	Applications of renewal theory (reliability, demand theory, etc.)

Keywords:

percolation process; phase transition

Citations:

Zbl 0647.60103