Homogenization and Percolation

The phenomenon of percolation can be conveniently modeled by a random structure of chess-board type, or random checkered tessellation. A structure of this kind is obtained if we split the plane into squares, painting each square, independently, black or white with probability p or 1 − p, respectively, where 0 ≤ p ≤ 1. Then the union of all black squares forms a random set F. Any two black squares are thought of as neighboring, or linked, if they have a common side or a vertex. In accordance with this arrangement, a finite number of black squares is said to form a path if these squares can be enumerated in such a way that any two consecutive numbers correspond to neighboring or linked squares. A set K consisting of black squares is connected if for any pair of squares belonging to K there exists a path containing these squares and belonging to K. Obviously, the set F is a union of separate connected components, which are called clusters.