Percolation

A simple example of a phase transition studied extensively since the 1950s involves taking a square lattice and filling in at random a certain density of black cells. In the limit of infinite size, there is a discrete transition at a density of about 0.592746, with zero probability below the transition to find a connected "percolating" cluster of black cells spanning the lattice, and unit probability above. (For a triangular lattice the critical density is exactly 1/2.) One can also study directed percolation in which one takes account of the connectivity of cells only in one direction on the lattice. (Compare the probabilistic cellular automata on pages 325 and 591. Note that the evolution of such systems is also analogous to the process of applying transfer matrices in studies of spin systems like Ising models.)