Percolation

One-Dimensional Percolation Theory Cont.

Now let us answer the question of what is the mean size of a cluster. An arbitrary site has a probability n_ss of being a part of a cluster of size s. The probability that a site is in a cluster of any size can be written as Σn_ss. We can therefore define w_s = n_s/Σn_s is equal to the probability that a cluster to which an arbitrary occupied site belongs is of size s. The mean cluster size can then be defined as

S = (Σ w_s= Σn_s*s²) /( Σn_s*s) (Eqn. 3)

Let us calculate this explicity by performing the summations. Note that the denominator is equal to p from equation 2. The numerator is then

The last equality comes simply from recalling that for a 1-D lattice p_c is one. So we have solved for the mean cluster size S. At this point, I should clarify a confusing point. An astute student may point out that if an infinite cluster exists, the summations above diverge to infinity. This is a valid complaint, and in truth, the above sums are only valid for p < p_c

Notice an interesting property of the mean cluster size. As p approaches p_c, then S diverges to infinity. How can we make sense of this strange phenomenon. Well, if an infinite cluster exists for a value of p at or above the percolation threshold, then just below, we should expect there to be clusters that are not quite infinite, but very large. These large clusters are what make S diverge.

Now we are going to define a correlation function g(r) as the probability that a site a distance r away from an occupied site is both occupied and in the same cluster. For one dimension, we can calculate this directly. The easiest example is for r=1. In this case, we are just asking, what is the probability a site next to an occupied site is also occupied. This is simple p. For a site of distance r away, there must be r sites in a line all occupied. Therefore, we have

g(r) = p^r (Eqn. 4)

For p < 1, the correlation function goes to zero exponentially as r goes to infinity. To reflect this, we can re-write the correlation function

g(r) = exp(-r/ξ) (Eqn. 5) where we have defined ξ as ξ = -1/ln(p) = 1/(p_c-p) (Eqn. 6)

ξ is the correlation length. It is characteristic distance over which the correlation function exponentially decays. Correlation lengths are ubiquitous in physics.

The last expression is actually an approximation for p close to p_c. This approximation is okay for us, as we are normally only interested in the behavior of the system near p_c. Notice that this approximation correctly maintains the divergence of ξ at p_c. Because of this divergence, behavior near p_c is often referred to as the asymptotic limit. Notice that in the asymptotic limit (i.e. when p is close to p_c) the average cluseter size S is proportional to ξ:

S ∝ ξ (p→p_c) (Eqn. 7) This last result does not generalize to higher dimensions. The more general relation between the mean cluster size and the correlation function is S= Σ g(r) (Eqn. 8)

where the summation is over all values of r.