Site Percolation

In discrete percolation theory, site percolation is a percolation model on a regular point lattice  in -dimensional Euclidean space which considers the lattice vertices as the relevant entities (left figure). The precise mathematical construction for the Bernoulli version of site percolation is as follows.

First, designate each vertex of  to be independently "open" with probability  and closed otherwise. Next, define an open path to be any path in  all of whose vertices are open, and define at the vertex  the so-called open cluster  to be the set of all vertices which may be attained following only open paths from . Write . The main objects of study in the site percolation model are then the percolation probability

(1)

and the critical probability

(2)

where here,  is defined to be the product measure

(3)

 is the Bernoulli measure which assigns  whenever  is closed and assigns  when  is open, and  is the percolation threshold. Site models for which  will have infinite connected components (i.e., percolations) whereas those for which  will not.

In general, site percolation is considered more general than bond percolation due to the fact that every bond model may be reformulated as a site model on a different lattice but not vice versa. Mixed percolation is considered to be a bridge between the two. Note, too, the existence of several other variants of site percolation; for example, one could drop the assumption of independence to obtain a non-Bernoulli, dependent site model.

REFERENCES

Chayes, L. and Schonmann, R. H. "Mixed Percolation as a Bridge Between Site and Bond Percolation." Ann. Appl. Probab. 10, 1182-1196, 2000.

Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.Hammersley, J. M. "A Generalization of McDiarmid's Theorem for Mixed Bernoulli Percolation." Math. Proc. Camb. Phil. Soc. 88, 167-170, 1980.