Percolation

Introduction

Consider a lattice of squares, in which every square or “site” can be in one of two states, “occupied” or “empty”. Each site is occupied independently of its neighbors. The probability that a site is occupied is p. This gives a model of site percolation.  Two or more occupied sites form a cluster if they are linked by a path of nearest neighbor connections between occupied sites. The picture below on the left shows a cluster of two occupied sites (filled squares).  The picture on the right shows two occupied sites that are not nearest neighbors and hence do not belong to the same cluster.

A procedure to study percolation is to generate a random number and then occupy a site if the random number is less than p. This is done for every site in the lattice. If p is small, then we expect only small isolated clusters to be present. On the other hand, if p is close to 1, we expect most of the occupied sites to be in a single large cluster that extends from one end of the lattice to the other. Such a cluster is said to span the lattice and is called a spanning cluster.

The figure below shows examples of site percolation clusters for p = 0.2, 0.4, 0.6, and 0.8. Spanning clusters are shown in red or blue.

The most important property of site percolation is that, in the limit of an infinite lattice, there exists a “threshold” probability, p_c, such that:

            For p ≥ p_c, one spanning cluster exists.

            For p < p_c, no spanning cluster exists and all clusters are finite.

The intrinsic characteristic of percolation is connectedness. The connectedness exhibits a qualitative change in behavior at a well-defined value of p. Hence the transition from a state with no spanning cluster to a state with one spanning cluster is a type of phase transition.

Site percolation is a simple model that is applicable to a number of physical processes including 1) electrical conductivity of a mixture of metallic and insulating materials, 2) the behavior of magnets diluted by non-magnetic materials, 3) the characterization of gels (e.g. why is jello different from broth?). It is also useful for understanding how diseases spread in a population, how oil flows through porous rock, and the spread of forest fires.

The Percolation Threshold

To find the percolation threshold we first choose a value for the number of sites on a side of the lattice, L. We then find, by numerical simulation, the probability, F(p), that there is a spanning cluster for occupation probability p. We next determine the value of p for which F(p) = 0.5. Since this depends on L, denote it by p_c(L). Finally we extrapolate to the limit  to get the percolation threshold, p_c.

The figure below shows F(p) for different L values.

For each value of p, 1000 lattices were produced and F(p) determined from the fraction that have a vertically spanning cluster. Because of fluctuations, p_c(L) is found by fitting a smooth function to F(p). A suitable function is

(1.1)

 Clearly when p = p_c(L), G(p) = 0.5. The figure below shows the results of such a fit.

To do the extrapolation to the limit , we fit a polynomial in 1/L to p_c(L). The constant is an estimate of p_c.

We find p_c = 0.5927, which is the correct value for the infinite square lattice.

Percolation and Critical Phenomena

Phase transitions that are associated with changes in temperature, such as melting or the transition from a ferromagnetic state to a paramagnetic state as the Curie temperature is passed, are thermodynamic phase transitions. Many properties of the geometric phase transition associated with percolation are similar to those of thermodynamic phase transitions. To illustrate some of these similarities, consider  which is defined by

(2.1)

For an infinite lattice,  = 0 for p < p_c and  = 1 for p = 1.

The figure below shows  for L = 128.

Note the similarity to the behavior of the magnetization of an Ising spin system with temperature.  does not go exactly to zero as  because of the finite size of the lattice.

Another quantity of interest is the mean cluster size S(p), which is defined by

(2.2)

 where s_n is the number of occupied sites in cluster n.  The spanning cluster is excluded from the sums. The figure below shows the results for L = 128.

Note the similarity to the Ising model specific heat. For the infinite lattice, there is a singularity at p = p_c and on either side of the singularity , where  is a critical exponent.

In the figure below we plot S(p) against L for p = 0.5, 0.59, and 0.7.

We see that for p = 0.5, and 0.7,  as L becomes large but for p = 0.59, S(p) continues to grow with L. What this means is that for p far from p_c, the properties of the system are indistinguishable from a truly macroscopic system. If p is close to p_c, then the length scale associated with a typical cluster is of order L and the behavior differs from a macroscopic system. In other words, finite size effects are never negligible at p_c.

Return to Class Notes for PHYS306