Like so many other problems in theoretical physics, the percolation problem can be solved exactly in one dimension, and some aspects of that solution seem to be valid also for higher dimensional cases.
 
 

Problem and System:

• Site percolation on an infinitely long linear chain
• `lattice' sites are placed in fixed distances.
• Each of these lattice sites is randomly occupied with probability p.

Definition of a cluster in 1-dim:
 

• A  group of neighboring occupied sites containing no empty site in between.

• A single empty site splits the group into two different clusters.

• In order that the cluster is separated from the other clusters, the site neighboring the left end of the cluster must be empty; and the same is true for the right end of the cluster.

Status of sites:
 

• All five central sites occupied.

• Two neighbors at both ends empty.

 

Probability of this configuration:
 

The probability of each site being occupied is p.
Since all sites are occupied randomly, the probability of two arbitrary sites being occupied is p², for three being occupied is p³, and for five being occupied is p⁵. (This product property of the probabilities is valid only for statistically independent events, as for random percolation).

The probability of one end having an empty neighbor is (1-p), and the probability for both ends empty is (1-p)² as the two ends are statistically independent.

Total probability of this configuration:

p⁵(1-p)².

 

Cluster number per lattice site:
 

If the total chain length is L, with L -> infinity,   then the total number of five-clusters, apart from effects from the chain ends, is:
 

L p⁵(1-p)²

We see that it is practical to talk about the number of clusters per lattice site, which is the total number divided by L and thus
 

n₅ = p⁵(1-p)²

This normalized cluster number is thus independent of the lattice size L and equals the probability that a fixed site is the end of a cluster.
 

We define n_s as the number of such s-clusters per lattice site:
 

n_s = p^s (1-p)²

This normalized cluster number is crucial for many of our discussions in two or three dimensions. It equals the probability, in an infinite chain, of an arbitrary site being the left hand end of the cluster.

For p<1, the cluster number goes exponentially to zero if the cluster size s goes to infinity.

Percolation threshold
 

For p= 1, all sites of the chain are occupied, and the whole chain constitutes one single cluster.

For every p smaller than unity, there will be some holes in the chain where a site is not occupied, which means that there is no continuous row of occupied sites, i.e., no one-dimensional cluster, connecting the two ends.

In other words, there is no percolating cluster for p below unity. Thus the percolation threshold is unity:

p_c = 1

We have learned that:
 

P + sum _{s = 1, 2, 3, ...}  n_s s = p.

For p < pc, P=0, therefore:
 

 sum _{s = 1, 2, 3, ...}  n_s s = p

This law can also be checked directly from expression for n_s = p^s (1-p)² and the formula for the geometric series:

sum_{s=1 -> infinity} p^s (1-p)² s =

(1-p)² sum_{s=1 -> infinity} p d(p^s)/dp =

(1-p)² p [ d (sum_{s=1 -> infinity} p^s) ] / dp =

(1-p)² p [ d (p/(1-p)) ] / dp = p
 

For higher dimension the above equation is also valid except that one has to take into account the sites in the infinite cluster separately, if one does not include them in the sum over all cluster sizes. There this equation  is restricted to p < p_c

Even in one dimension at p = p_c = 1 there is only one cluster covering the whole lattice. Thus s=infinite value and n_s = 0, which makes the equation undefined at p=1.

Average cluster size

We have defined cluster size S as:
 

S = ( sum _{s = 1 to infinity}  n_s s² ) / ( sum _{s = 1 to infinity}  n_s s )

Let us now calculate this mean cluster size explicitly:
 

The denominator is simply p, as shown above.

The numerator is, by substituting n_s = p^s (1-p)²:
 

(1-p)² sum _{s = 1 to infinity}  ( s²  p^s ) =

(1-p)² (p d/dp)²  sum _{s = 1 to infinity}  ( p^s )

where the trick from our previous derivation is applied twice in order to calculate sum by using suitable derivatives of easier sums.

Thus
 

S = (1 + p) /  (1 - p)        (p < p_c)

The mean cluster size diverges if we approach the percolation threshold. We will obtain similar results in more than one dimension. This divergence is very plausible, for if there is an infinite cluster present above the percolation threshold, then slightly below the threshold one already has very large (though finite) clusters. Thus a suitable average over these sizes is also getting very large, if one is only slightly below the threshold.
 

Correlation function
 

Correlation function or pair connectivity g(r) has been defined as the probability that a site a distance r apart from an occupied site belongs to the same cluster.

For r=0 that probability g(0) equals unity.

For r=1 the neighboring site belongs to the same cluster if and only if it is occupied; this is the case with probability p.

For a site at distance r, this site and the (r-1) sites in between this site and the origin at r=0 must be occupied without exception, which happens with probability pr. Thus:
 

g(r) = p^r        for all p and r.

For p<1 this correlation function goes to zero exponentially if the distance r goes to infinity:
 

g(r) = exp(-r/xi)

where
 

xi = - 1/ln p ~ 1/(p_c - p)

The last equality in the above equation is valid only for p close to p_c = 1 and uses the expansion ln(1-x) = - x for small x.

The quantity xi is called the correlation length and we see that it also diverges at the threshold.

We will see in higher dimensions that the correlation length is proportional to a typical cluster diameter.

This relation is quite obvious here. The length of a cluster with s sites is (s-1), not much different form s if s is large. Thus the average length xi varies as the average cluster size S:
 

S  ~ A xi, (p -> p_c).

where A is a constant.

Unfortunately, this relation becomes more complicated in higher dimensions. Rather more generally valid is a relation between the sum over all distances r of the correlation function, and the mean cluster size:

sum_r g(r) = S.
 

Beyond 1-dim system:
 

• In one dimensional systems, certain quantities diverge at the percolation threshold, and that the divergence can be described by simple power laws like 1/(p_c - p), at least asymptotically close to p_c. The same seems true in higher dimensions where the problems have not been solved exactly.

• The quantities S and xi have counterparts in higher dimensions for thermal phase transitions. In fluids near their critical point, critical opalescence is observed in light-scattering experiments, since the compressibility (analogous to S) and the correlation length xi diverge there.

• One may utilize one-dimensional percolation further by calculating the cluster numbers in finite one-dimensional chains. Then one can check the general concepts of finite-size scaling and universality.

• The one-dimensional case is now solved exactly, whereas for the d-dimensional case only small clusters will be treated exactly. There is another case with an exactly known solution, the Bethe lattice.

Homework