Percolation arises from a number of problems in condensed matter physics. These problems share common features.
 

Examples:
 
 

• Example 1: A mixture of metallic and non-metallic material with a voltage applied across it.

Figure 4-1

Question:

Given the voltage, what is the current passing through the sample?

We imagine that the mixture consists of small grains. For simplicity, we model the material as small conducting segments distributed among non-conducting background. We also assume that the conducting segments can only sit on a regular lattice, e.g., a two-dimensional square lattice.

Our question is how physical quantity like current or conductivity varies with the concentration of the conducting component.

The current as a function of the fraction p of conducting bonds will qualitatively look like the Figure shown below:

Figure 4-2

• Example 2:  Dilute magnet at low temperatures.
A ferromagnetic material shows magnetization at low enough temperatures. When the material is diluted with nonmagnetic material, the magnetization may disappear.

Here the question is how magnetization depends on the concentration of dilution.
 
 
 

• Example 3: Fluid passing through a porous material.

The porous media (e.g., rock) have many small random channels.

Figure 4-3

The question is whether the fluid can flow through the material. The answer, of course, depends on the concentration and nature of the channels. If there are few open short channels distributed at random, the fluid cannot pass through, or in other words, the system does not percolate. When more channels are there, the fluid can pass through the solid -- we say that the system is percolating.

At some particular concentration the system changes its behavior, that particular concentration or situation is called percolation threshold.

How to deal with percolation problems?
 

• If you cannot deal with it, simplify it.

The spirit of physics (or perhaps more prevailing in condensed matter physics) is to simplify the problem as far as possible, retaining only the essential features.
 

• Focus on important things.

A mathematical model for percolation can be built which takes into account only the most important features, neglecting non-essential details.
 

• Watch out for universal behavior

It turns out that certain aspects of the problem do not depend on the details. This is quite remarkable and is called universality.
 
 

Site percolation
 

Let us consider a two-dimensional problem. Three or higher dimensional cases are similar.

Mathematical model:

Consider a system of two-dimensional square lattice. For computer simulation, we also work on a finite lattice. Let's take the size of the system to be L X L. Each site may be occupied or empty. We may denote this by an occupation number:
 

n(i,j) = 0, or 1

where 0 < i,j < L specifies the location of a lattice site.

Figure 4-4

The occupation number zero indicates an empty site and one for an occupied site. The collection  n(i,j)  denotes one particular configuration.
 

Computer algorithm:

In a computer program, we can use a two-dimensional array taking values zero and one to represent a configuration. Since only lattice sites are involved, this is called a site percolation model.

How do we decide which sites to occupy? It is simple. We decide at random. More precisely, we go through each site once and occupy the site with probability p, and consequently, leave the site empty with probability 1-p.

In bond percolation, instead of sites, bonds are occupied with probability p. A bond connects two neighboring sites.  The mathematical model is similar to the one above. The figure below is a bond percolation configuration.

Figure 4-5

Statistical description

Since probability enters into our description of the site (or bond) percolation, our problem is statistical in nature.

Site percolation as an illustration:

• What is the probability that a particular configuration n(i,j) appears?

Since each appearance of an occupied site has probability p independent of other sites, each appearance of an empty site has probability 1-p, the whole configuration appears with probability
 

p^m ( 1- p)^{Ld - m}

where d=2 is the dimension, m = sum n(i,j) is the total number of occupied sites, and Ld = L^d.

The average of certain quantity A ({n(i,j)}) which depends on the configuration means:
 

<A> = sum _{all conf.} A({n(i,j)}) p^m (1-p)^Ld-m

 

where the summation is over all possible configurations.

There are a total of 2^Ld of configurations. The average can be estimated approximately by Monte Carlo method.

Percolation Threshold
 

Percolation threshold P_c is defined as the value of probability p below which there are only finite size clusters in an infinitely large system, above which there is at least one cluster which is infinitely large.
 

Figure 4-6

A cluster is defined as a set of sites which are connected through nearest neighbors. The size s of a cluster is the number of sites in the cluster.

Percolation threshold value depends on the type of lattice and dimensions. Here are selected percolation thresholds for various lattices . `Site' refers to site percolation and `bond' to bond percolation. Some of the results are exact, others are obtained approximately.
 

 Important Physical Quantities in Percolation
 
 

Clusters

The definition of a cluster in a configuration is somewhat arbitrary. But the following definition is commonly used:

A cluster is a collection of sites connected by bonds (for bond percolation) or a collection of nearest neighbor sites (for site percolation).

When the sites are at the boundary, we wrap the lattice (like a torus) to decide connectivity. The size s of a cluster is the number of sites in the cluster.

Here is a site percolation configuration and clusters:
 

Cluster number n_s
 

It is probably the most important quantity conceptually since many other quantities can be derived from it.

The cluster number n_s is the  average number of clusters having exactly size s, normalized by the system size L^d.

The cluster number n_s depends on the probability p as well as system size L. But the L-dependence quickly approaches a limit as L goes to infinity.

Most often, we focus on the properties of n_s in the infinity lattice size limit.

In this case, we consider a very large system. This is not something artificial. This is physical. The mathematical sites in our percolation model represent atoms or groups of atoms. Since the typical size of atoms or molecules is of order 10^-8 centimeter, and typical size of a laboratory sample is of order of few centimeter. So, there are typically 10²³ atoms, which is very large.

 

This is also called order parameter of the percolation problem or strength of the infinite network.

A note about labeling:

The lowercase letter p denotes the probability of occupying a site (concentration of occupied sites). We'll use the uppercase P to denote  percolation probability.

Definition of percolation probability

P is defined as the probability that the origin or any other arbitrarily selected site belongs to an infinite cluster. The infinite cluster is called a percolating cluster since it spans the whole lattice.

General behavior:

For p < p_c,  P = 0,

because there exists no infinite cluster.

For p > p_c, the quantity P becomes positive, and approaches 1 as p -> 1.
 

Relationship between  n_s, p, and P:

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

The equation can be understood as follows:

A site has probability p being occupied; if it is occupied, it is either on the infinitely large cluster or on a finite size cluster.
The probability that the site belongs to the infinite cluster is P.
The probability on a finite cluster of size s is ns s.

The later probability can be understood this way -- we have Ld ns clusters of size s; out of total of Ld lattice sites, there are Ld n_s s sites belonging to clusters of size s. If we pick a site at random, we have the probability n_s s picking the sites belonging to a cluster of size s. The probability of the site being occupied is p and is the sum of probabilities on the infinite cluster and on finite clusters of size s=1, 2, 3, ...

 

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

The summation runs to infinity. But it does not include contribution from infinitely large percolating cluster.

Pair correlation function g(r) measures how two points are related. It is defined as the probability that a site distance r away from an occupied site belongs to the same cluster. This quantity is related to S by:

sum _r g(r) = S ,  (p < p_c).