Percolation

Percolation is a geometric analog of thermal phase transitions that is interesting on its own. Consider a square lattice, large enough (ideally infinite) so that we may ignore boundary effects in our discussion. Let each site of the lattice be empty or occupied with a probablity $p$: The occupation of the sites is decided by a random process, so the sites are independent of each other. Now for $p=0$, all sites are empty while for $p=1$ all sites are occupied. Define a cluster as a group of nearest neighbour sites that are occupied. As $p$ increases from $0$, a critical point $p=p_c$ is reached when a large cluster is formed stretching from one edge of the lattice to the opposite edge. The value $p_c$ is called the percolation threshold and at this point there is a significant change in the properties of the lattice. For example, if sites represent pores in a rock, and being occupied means the pores are open, then at the percolation threshold water can seep through from one end of the rock to the other. There are other physical problems that can be studied with a percolation model, such as forest fires, or conductivity of a random network. Since percolation is a random process each simulation on a lattice for fixed $p$ will give rise to different clusters of varied sizes and one must discuss statistical properties of relevant quantities (such as cluster size) obtained after an averaging. It is found that near the percolation threshold the physically interesting quantities diverge and show power-law behaviour similar to that near the crtical point of a second-order phase transition, with $p$ playing the role analogous to temperature. Therefore for the percolation problem one can again define critical exponents and show their universality (that is, independence from underlying lattice type). At the critical point, the structure of the clusters becomes fractal, that is, there are clusters of all scales and the self-similarity dimension is fractal. This is perhaps not very surprising since at the critical point the properties of the system become scale-invariant and obey power-laws.