What is a critical behavior?
 

In layman's term:

• A drastic change following small variation of one or more parameters.

Examples:

Phase transition of water:

• liquid <-> ice transition at freezing temperature.

 

Percolation:

Qualitative behaviors of the percolation probability P and average cluster size S:

 

An interesting thing happens at the percolation threshold, pc; this point is also referred as critical point, and the phenomena associated with it are called critical phenomena.

The modern theory of critical phenomena is a well-developed theory and a great success of statistical mechanics. K. Wilson won his Nobel Prize for his renormalization group theory of critical phenomena.

Critical phenomena will be studied again in molecular dynamics and Monte Carlo methods later in the course.

Critical Exponents:
 

Near the critical point pc the two quantities can be described empirically as:
 

            0 ,                   p < p_c;
P ~  {
            (p-p_c)^beta,    p > p_c, p -> p_c.
 
 

S  ~ | p - p_c |^gamma,   p -> p_c.

 

Where beta and gamma are some constants called critical exponents.

The proportionality sign ~  takes the usual meaning:
 

The ratio of the left-hand side and the right-hand side approaches a constant as p approaches p_c.
 

• Critical behavior only occurs near a critical point:
 

The equation or laws that the quantity P or S obeys is called asymptotic law since it takes that form only when p is close to the critical value pc.

• Critical exponents like beta and gamma are universal.

More precisely, the value beta is the same for a whole class of models and experimental samples independent of the details of the models or samples. The value depends only on the dimensionality of the problem.

Two dimensional percolation is different from three dimensional percolation. It depends whether the percolation is short ranged or long ranged. But it does not depend whether the lattice is a square lattice or triangular lattice. The critical exponents are the same on both lattices.

The universality of exponents justifies the study of the simplest possible model, and the result also applies to more general situations. One of the tasks of this study is to calculate the critical exponents by computer simulation.

Scaling assumption of the cluster number
 

The percolation probability P and average cluster size S have a simple power-law behavior near the critical point pc. What about the cluster number n_s?

This quantity is more complicated because it depends on two variables, the cluster size s and occupation probability p. So we should write more precisely as n_s(p).

Except in one dimension and a special lattice (Bethe lattice), there are no exact solutions for the percolation problems. Based on many examples and experience in critical phenomena, the following functional form has been assumed:
 

n_s(p) = s^-tau f[(p-p_c)s^sigma],  p ->  p_c,  s -> infinity

Where tau and sigma are also critical exponents. The above equation is called the scaling assumption for the cluster number.

The exact values for the exponents tau, sigma and the exact form for the function f(z) are not known in general. They must be determined by (computer) experiment or by other methods. In fact, the actual assumption must be tested. People believe that the assumption indeed holds. There has been no rigorous proof of the assumption. Do you want to take on the challenge?
 
 

We have introduced four different exponents beta, gamma, tau, and sigma. Since P and S can be derived from n_s(p), the exponents are not independent. They are related. This is one of the consequence of the scaling assumption.

A test of the exponent equality is also a test for the scaling assumption. From the equation:
 

P = p - sum _{s = 1, 2, 3, ...} (n_s(p) s)

together with the scaling assumption, we have:
 

P = p - sum _{s=1 to infinity} s^1-tau f((p-p_c) s^sigma).

We'll use a trick to write the equation in the variable p-p_c. Since P=0 at p=pc, setting p=pc and subtracting it from the above equation, we have:

P = (p-p_c) - sum _{s=1 to infinity} s^1-tau [f((p-p_c) s^sigma) - f(0)].
 

We'll convert the summation to integral. The error introduced is small near the percolation threshold.

P = (p-p_c) - int _{s=s0 to infinity} s^1-tau [f((p-p_c) s^sigma) - f(0)]ds

Only the upper limit matters. The lower limit does not influence the finite result very much. So we set it at some arbitrary value s0. By changing the integration variable
 

s=(x/(p-p_c))^1/sigma

one finds:
 

P = (p-p_c) - ( p -p_c)^{(tau - 2)/sigma} C  ~  (p-p_c)^beta

where C is a constant.

Comparing the exponents, using the fact that beta < 1, (neglecting (p-p_c) term), we have the first relation:
 

beta =  tau - 2/sigma

Now we calculate the average cluster size. This quantity diverges at the percolation threshold. The divergence comes from the numerator:  sum (s2 ns ), since the denominator sum (s n_s)  simply approaches a constant p_c.

To leading order of the divergence term, we can approximate it by p_c and thus:
 
 
 

S =  sum s² n_s(p)  / sum s n_s(p)  ~  sum s² n_s(p) / p_c

   ~  int s² n_s(p) ds   ~   int  s^2-tau  f((p-p_c) s^sigma) ds

   ~  |p-p_c|^{(tau - 3)/sigma}  ~   |p-p_c|^-gamma

 

Again we have changed integration variable, and the value of the integral is just some number independent of p. Thus we have the second relation:
 

gamma =  3 - tau/sigma

Note that in order that both beta and gamma are positive we need 2 < tau < 3.

Here is a list of percolation exponents for d=2, 3, 4, 5, 6-e (e is a small value close to zero), and for the Bethe lattice:

The dimension d=6 is called upper critical dimension, above which the exponents become independent of the dimensionality and take the values of the Bethe lattice results.

The exponent nu is called correlation length exponent, defined below. And D is the fractal dimension of clusters at the percolation threshold.

Critical Exponents Related to Cluster Structure
 
 

Cluster perimeter
 
 

We'll consider site percolation. Each individual cluster is called a lattice animal. The empty sites immediately adjacent to the cluster (empty neighbors) are called perimeter sites.

 

Let's use:
 

• t as the number of perimeter sites of a given lattice animal
• s as the total number of occupied sites of the lattice animal.
• g_st to denote the number of lattice animals (cluster configurations) with size s and perimeter t

then:

 

n_s(p) = sum _t  [ g_st p^s ( 1-p)^t  ]

is the average number of s clusters per lattice site.

If we have an analytic formula for gst, then we should have solved the percolation problem exactly. Unfortunately, this is not the case except in one dimension. However, we can find ns for the first few small values of s.

For s=1 and dimension d=2:

n1 = p ( 1 - p)⁴
g_1,4 = 1, all other g_st = 0.

For s=2, d=2:

n2 = 2 p²(1-p)⁶
g_2,6 = 2, all other g_st = 0.

For s=3, d=2:

n3 = 2 p³ ( 1 - p)⁸ + 4 p³(1-p)⁷
g_3,8 = 2, g_3,7 = 4, all other g_st = 0.

 

The meaning of the formula correlates closely with the figures shown above . With the help from computer, people were able to obtain gst for s upto 30.

The average perimeter t of a cluster is not proportional to the surface area of the cluster. This is because there are lot of holes in a percolation cluster. If we have one hole for, say, every thirty sites we have a perimeter proportional to the number of sites. Thus:

t ~ s, s  -> infinity

This result can be proved more rigorously.

Correlation length

We have introduced the correlation function g(r), which is the probability that a site at distance r from a given occupied site is also occupied and belongs to the same cluster. The function behaves at large distance in the form

g(r) ~  e^-r/xi,  p < p_c.

For p>p_c it approaches a constant P². The quantity xi is called correlation length.

The correlation length can also be defined as:
 

xi² =  sum_r r² g(r)  / sum_r g(r)

The correlation length measures the typical size of clusters. Near the percolation threshold, clusters become very large, and the correlation length diverges,:
 

xi  ~  | p - p_c | ^-nu

where nu is called correlation length exponent.
 

Cluster radius and fractal dimension
 

The correlation length is a global quantity for the whole system. Cluster radius measures the dimension of clusters having exactly s sites. Since the clusters are rather irregular, we can only give an average quantity which measures the size of the clusters.

The clusters at the percolation threshold are fractals, in the sense that the total size s or mass (imagine each occupied site has certain weight) is related to its size as
 

s ~ R_s^D,  p = p_c, s  >>1

D is called the fractal dimension of the clusters. This fractal behavior for large but finite clusters also holds for the largest cluster in a finite system. If L is the linear size of the system with Ld lattice sites, the mass M (or total number of sites) of the largest cluster varies with the system size as:
 

M  ~ L^D

According to scaling theory, D can be related to other exponents as:
 

D = d -  beta/nu,   d < 6