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.