Infinite-Dimensional Percolation

Abstract

In this chapter we address the percolation problem on an infinite-dimensional lattice without loops. In this case, it is possible to calculate several of the properties of the percolation system analytically. This allows us to develop a general theory and to develop concepts to be used for finite-dimensional systems. We introduce the infinite-dimensional Bethe lattice for a given coordination number. We find an exact solution for P and the average cluster size S, and use a Taylor-expansion to find an expression for $n(s,p)$. The methods and functional forms for $n(s,p)$ we introduce here, are used to interpret results in finite dimensions.

In this chapter we address the percolation problem on an infinite-dimensional lattice without loops. In this case, it is possible to calculate several of the properties of the percolation system analytically. This allows us to develop a general theory and to develop concepts to be used for finite-dimensional systems. We introduce the infinite-dimensional Bethe lattice for a given coordination number. We find an exact solution for P and the average cluster size S, and use a Taylor-expansion to find an expression for $n(s,p)$. The methods and functional forms for $n(s,p)$ we introduce here, are used to interpret results in finite dimensions.

We have now seen how the percolation problem can be solved exactly for a one-dimensional system. However, in this case the percolation threshold is $p_c = 1$, and we were not able to address the behavior of the system for $p>p_c$. There is, however, another system in which many features of the percolation problem can be solved exactly. This is percolation on a regular tree structure on which there are no loops. The condition of no loops is essential. This is also why we call this system a system of infinite dimensions, because we need an infinite number of dimensions in Euclidean space in order to embed a tree without loops. In this section, we will provide an explicit solution to the percolation problem on a particular tree structure called the Bethe lattice [5].

The Bethe lattice, which is also called the Cayley tree, is a tree structure in which each node has Z neighbors. This structure has no loops. If we start from the central point and draw the lattice, the perimeter grows as fast as the bulk. Generally, we will call Z the coordination number. The Bethe lattice is illustrated in Fig. 3.1.

Fig. 3.1

Four generations of the Bethe lattice with coordination number $Z=3$. (a) Illustration of a two-dimensional embedding of the lattice. (b) Illustration of how a central node is connected to three branches

3.1 Percolation Threshold

If we start from the center in Fig. 3.1 and move along a branch, we find $(Z-1)$ new neighbors from each of the branches. To get a spanning cluster, we need to ensure that at least one of the $Z - 1$ sites are occupied on average. That is, the occupation probability, p, must be:

$$\displaystyle \begin{aligned} p(Z-1) \ge 1 \; , \end{aligned} $$

(3.1)

in order for this process to continue indefinitely.

We associate $p:c$ with the value for p where the cluster is on the verge of dying out, that is

$$\displaystyle \begin{aligned} p_c = \frac{1}{Z-1} \; . {} \end{aligned} $$

(3.2)

For $Z=2$ we regain the one-dimensional system, with percolation threshold $p_c = 1$. However, when $Z>2$, we obtain a finite percolation threshold, that is, $p_c < 1$, which means that we can observe the behavior both above and below $p:c$.

In the following, we will demonstrate how we can find the density of the spanning cluster, $P(p)$, and the average cluster size S, before we address the scaling behavior of the cluster number density $n(s,p)$.

3.2 Spanning Cluster

We will use a standard trick to find the density $P(p)$ of the spanning cluster when $p>p_c$. The technique is based on starting from a “central” site, and then address the probability that a given branch is connected to infinity.

Relating P to $n(s,p)$

We start by noting that P is the probability for a site to be connected to the spanning cluster. If a site is present, which has a probability p, it must either belong to the infinite cluster with probability P or to one of the finite clusters with probability $\sum _s s n(s,p)$. This means that

$$\displaystyle \begin{aligned} p = P + \sum_s s n(s,p) \; , \end{aligned} $$

(3.3)

The sum $\sum _s s n(s,p)$ is the probability that the site is part of a finite cluster, which means that it is not connected to infinity. We introduce $Q(p)$ to denote the probability that a branch does not lead to infinity. The concept of a central point and a branch is illustrated in Fig. 3.1.

Deriving an Equation for $Q(p)$

If the probability that a site is not connected to infinity in a particular direction is Q, then the probability that the site is not connected to infinity in any direction is $Q^Z$. The probability that the site is connected to infinity is therefore $1 - Q^Z$. In addition, we need to include the probability p that the site is occupied. The probability that a given site is connected to infinity, that is, that it is part of the spanning cluster, is therefore

$$\displaystyle \begin{aligned} P = p(1-Q^Z) \; . \end{aligned} $$

(3.4)

Now, we need to find an expression for $Q(p)$. We will determine Q through a consistency equation. Let us assume that we are moving along a branch, and that we have come to a point k. Then, Q gives the probability that this branch does not lead to infinity. This can occur either if site k is not occupied, which has a probability $(1 -p)$, or if site k is occupied, which has probability p, and all of the $Z-1$ branches that lead out of k are not connected to infinity, which has probability $Q^{Z-1}$. The probability Q for the branch not to be connected to infinity is therefore

$$\displaystyle \begin{aligned} Q = (1-p) + pQ^{Z-1} \; . {} \end{aligned} $$

(3.5)

We check this equation for the case $Z=2$, which corresponds to a one-dimensional system. In this case we have $Q = 1 - p + pQ$, which gives, $(1-p)Q = (1-p)$, where we see that when $p\neq 1$, $Q=1$. That is, when $p<1$ all branches are not connected to infinity, implying that there is no spanning cluster. We therefore regain the results from one-dimensional percolation theory.

Solving to Find $Q(p)$

We could solve this equation for general Z. However, for simplicity we will restrict ourselves to $Z=3$, which is the smallest Z that gives a behavior different from the one-dimensional system. We recall from (3.2) that for $Z=3$, $p_c = 1/2$. We insert $Z=3$ in (3.5), which gives

$$\displaystyle \begin{aligned} Q = 1 - p + pQ^2 \; , \end{aligned} $$

(3.6)

$$\displaystyle \begin{aligned} pQ^2 - Q + 1 - p = 0 \; . \end{aligned} $$

(3.7)

The solution of this second order equation is

$$\displaystyle \begin{aligned} Q = \frac{+1 \pm \sqrt{1 - 4p(1-p)}}{2p} = \frac{1 \pm \sqrt{(2p - 1)^2}}{2p} = \frac{1 \pm | (2p - 1) |}{2p} \; . \end{aligned} $$

(3.8)

First, we notice that for $(2p-1)=0$, that is for $p=1/2$ (which is $p:c$ for $Z=3$), we find that $Q = 1/2p = 1$. Therefore, no branch propagates to infinity for $p=p_c = 1/2$. Second, for $(2p-1)<0$, which corresponds to $p<1/2=p:c$, we get the solution

$$\displaystyle \begin{aligned} Q = \frac{1 \pm (1-2p)}{2p} \; , \end{aligned} $$

(3.9)

which has two solutions: $Q = 1$ or $Q = (1-p)/p$. The second solution $(1-p)/p$ is greater than 1 when $p<1/2$. We therefore conclude that for $p<1/2$ and for $p=1/2$, no branch propagates to infinity. This means that there is no infinite cluster (no spanning cluster) for $p \le 1/2 = p_c$. Third, for $(2p-1)>0$, which corresponds to $p>1/2=p_c$, we get the solution

$$\displaystyle \begin{aligned} Q = \frac{1 \pm (2p-1)}{2p} \; , \end{aligned} $$

(3.10)

which has two solutions: $Q=1$ or $Q = (1-p)/p$. The second solution is smaller than 1 when $p>1/2$. This means that there is a finite probability for a branch to propagate to infinity and for there to be a spanning cluster. We have therefore found that for $p \le 1/2$, there is no spanning cluster, but for $p>1/2$ there is a finite probability for a spanning cluster. This finding confirms that $1/2$ indeed is the percolation threshold for this system.

Finding $P(p)$

We insert $Q=(1-p)/p$ back into the equation for $P(p)$ to find the behavior of $P(p)$ for $p>p_c=1/2$:

$$\displaystyle \begin{aligned} P = p(1 - Q^3) = p(1 - (\frac{1-p}{p})^3) \; . {} \end{aligned} $$

(3.11)

This result is illustrated in Fig. 3.2. We notice that when $p \rightarrow 1$, we have that $(1-p)/p \rightarrow 0$. Therefore $P \propto p$ in this limit. To address the behavior when $p \rightarrow p_c = 1/2$, we use the identity $(1-a^3) = (1-a)(1+a+a^2)$ and that $1-(1-p)/p = 2(p-1/2)$ to rewrite the expression in (3.11) to become

$$\displaystyle \begin{aligned} P = p(1 - (\frac{1-p}{p})^3) = p \left(1 - \frac{1-p}{p} \right) \left(1 + \frac{1-p}{p} + \left(\frac{1-p}{p}\right)^2 \right) \; . \end{aligned} $$

(3.12)

Fig. 3.2

(Top) A plot of $P(p)$ as a function of p for the Bethe lattice with $Z=3$. The tangent at $p = p_c$ is illustrated by a straight line. (Bottom) A plot of the average cluster size, $S(p)$, as a function of p for the Bethe lattice with $Z = 3$. The average cluster size diverges when $p \rightarrow p_c=1/2$ both from below and above

We can rewrite this as

$$\displaystyle \begin{aligned} P = 2\left(p - \frac{1}{2}\right) \left(1 + \frac{1-p}{p} + \left(\frac{1-p}{p}\right)^2 \right) \; . {} \end{aligned} $$

(3.13)

We Taylor expand the second term around $p = p_c = 1/2$:

$$\displaystyle \begin{aligned} \left(1 + \frac{1-p}{p} + \left(\frac{1-p}{p}\right)^2 \right) \simeq 3 + \mathcal{O}(p-p_c) \; , \end{aligned} $$

(3.14)

which when inserted back into (3.13) gives

$$\displaystyle \begin{aligned} P \simeq 6 (p - p_c) + \mathcal{O}\left((p-p_c)^2\right) \; . \end{aligned} $$

(3.15)

Since we are only interested in the leading term, we have found that for $p>p_c = 1/2$ we can approximate $P(p)$ as:

$$\displaystyle \begin{aligned} P(p) \simeq B (p - p_c)^{\beta} \; , \end{aligned} $$

(3.16)

where $B=6$ and the exponent $\beta = 1$. The density of the spanning cluster, P, is therefore a power-law in $(p - p_c)$ with exponent $\beta$. In general, this exponent depends on the dimensionality of the lattice, but it does not depend on lattice details, such as the number of neighbors Z. We will leave it as an exercise for the reader to show that $\beta$ is the same for $Z = 4$.

The approach we have used here is often called a mean field solution or a self-consistency solution: We assume that we know Q, and then solve to find Q. We will use similar methods later.

3.3 Average Cluster Size

We will use a similar method to find the average cluster size, $S(p)$. We start by defining $T(p)$ as the average number of sites connected to a given site on a specific branch, such as in branch 1 in Fig. 3.1. The average cluster size S is then given as

$$\displaystyle \begin{aligned} S = 1 + ZT \; , {} \end{aligned} $$

(3.17)

where the 1 represents the central point, T is the average number of sites on each branch and Z is the number of branches. We will again attempt to find a self-consistent solution for T, starting from a center site. The average cluster size T is found from summing the probability that the next site k is empty, $(1-p)$, multiplied with the contribution to the average, in this case (0), plus the probability that the next site is occupied, p, multiplied with the contribution in this case, which is the contribution from the site (1) and the contribution of the remaining $Z-1$ subbranches. In total:

$$\displaystyle \begin{aligned} T = (1-p)\, 0 + p \, (1 + (Z-1)T) \; , \end{aligned} $$

(3.18)

We solve for T, finding

$$\displaystyle \begin{aligned} T = \frac{p}{1 - p(Z-1)} \; . \end{aligned} $$

(3.19)

This expression diverges when $1-p(Z-1)=0$, that is, for $p = 1/(Z-1)$, which we recognize as $p_c = 1/(Z-1)$. We insert this in (3.17) as find that S is

$$\displaystyle \begin{aligned} S = 1 + ZT = \frac{1 + p}{1 - (Z-1)p} = \frac{1+p}{1-\frac{p}{p_c}} = \frac{p_c(1+p)}{p_c - p} \; , \end{aligned} $$

(3.20)

which is illustrated in Fig. 3.2. This is a special case for $S(p)$, which in general can be written on the general form

$$\displaystyle \begin{aligned} S = \frac{\varGamma}{(p_c - p)^{\gamma}} \; . \end{aligned} $$

(3.21)

For the Bethe lattice with coordination number Z, we have found that $p_c = 1/(Z-1)$ and that the exponent is $\gamma = 1$. where our argument determines $p_c = 1/(Z-1)$, and the exponent $\gamma = 1$. The average cluster size S therefore diverges as a power-law when p approaches $p:c$. The exponent $\gamma$ characterizes the behavior. This is a general results. The value of $\gamma$ depends on the dimensionality, but not on the details of the lattice. Here, we notice that $\gamma$ does indeed not depend on Z.

3.4 Cluster Number Density

In order to find the cluster number density for the Bethe lattice, we start by developing a more general way to find the cluster number density. To find the cluster number density for a given s, we need to find all possible configurations, $c(s)$, of clusters of size s, and sum up their probability:

$$\displaystyle \begin{aligned} n(s,p) = \sum_{c(s)} p^s (1-p)^{t(c)} \; . \end{aligned} $$

(3.22)

This was simple in one dimension, because there was only one possible configuration for a given s. Just like in one dimension we include the term $p^s$, because we know that we must have all the s sites of the cluster present. In addition, we need to include a term that takes into account that all the neighboring sites must be empty. In general, we introduce the notation $t(c)$ for the number of neighbors for configuration c. In one dimension, t is always 2, but in higher dimensions different clusters may have different numbers of neighbors, as illustrated for the case of a two-dimensional system in Fig. 3.3. We therefore need to include the term $(1-p)^t$ to account for the probability for the t neighbors to be unoccupied. Based on this, we realize that we may sum over all values of t. However, we then need to include the effect that there are several clusters that can have the same t. We will then have to introduce a degeneracy factor $g:{s,t}$ which gives the number of different clusters that have size s and a number of neighbors equal to t. The cluster number density can then be written as

$$\displaystyle \begin{aligned} n(s,p) = p^s \sum_t g_{s,t} (1-p)^{t}\; . \end{aligned} $$

(3.23)

Fig. 3.3

Illustration of the 6 possible configurations for a two-dimensional cluster of size $s=3$

Degeneracy for Two-Dimensional Clusters

We can illustrate these concept for two-dimensional percolation. Let us study the case when $s = 3$. In this case there are 6 possible clusters for size $s = 3$, as illustrated in Fig. 3.3. There are two clusters with $t=8$, and four clusters with $t=7$. There are no other clusters of size $s=3$. We can therefore conclude that for the two-dimensional lattice, we have $g:{3,8}=2$, and $g_{3,7} = 4$, and $g_{3,t} = 0$ for all other values of t.

Degeneracy for the Bethe Lattice

For the Bethe lattice, there is a particularly simple relation between the number of sites, s, and the number of neighbors, t. We can see this by looking at the first few generations of a Bethe lattice grown from a central seed. For $s = 1$, the number of neighbors is $t_1 = Z$. To add one more site, we have to remove one neighbor from what we had previously, and then we add $Z-1$ new neighbors, that is, for $s=2$ we have $t_2 = t_1 + (Z-2)$. Consequently we get an iterative relation for the number of neighbors $t:k$ when we have k sites:

$$\displaystyle \begin{aligned} t_k = t_{k-1} + (Z-2) \; , \end{aligned} $$

(3.24)

and therefore:

$$\displaystyle \begin{aligned} t_s = s(Z-2) + 2 \; . \end{aligned} $$

(3.25)

Cluster Number Density

The cluster number density, given by the sum over all t, is therefore reduced to only a single term for the Bethe lattice

$$\displaystyle \begin{aligned} n(s,p) = g_{s,t_s} p^s (1-p)^{t_s} \; , \end{aligned} $$

(3.26)

For simplicity, we will write $g_s = g_{s,t_s}$. In general, we do not know $g:s$, but we will show that we still can learn quite a lot about the behavior of $n(s,p)$. The cluster density can therefore be written as

$$\displaystyle \begin{aligned} n(s,p) = g_s p^s (1-p)^{ 2 + (Z-2)s} \; . \end{aligned} $$

(3.27)

We rewrite this as a common factor to the power s:

$$\displaystyle \begin{aligned} n(s,p) = g_s [p(1-p)^{Z-2}]^s (1-p)^2 \; , {} \end{aligned} $$

(3.28)

which in the special case for $Z = 3$, which we studied above, becomes

$$\displaystyle \begin{aligned} n(s,p) = g_s [ p (1-p) ]^s (1-p)^2 \; . \end{aligned} $$

(3.29)

Taylor Expansion Around $p=p:c$

Let us now address $n(s,p)$ for p close to $p:c$ for a general value of Z. In this range, we will do a Taylor expansion of the term $f(p) = p(1-p)^{Z-2}$, which is raised to the power s in the equation for $n(s,p)$ in (3.28). The shape of $f(p)$ as a function of p is shown in Fig. 3.4. The maximum of $f(p)$ occurs for $p = p_c = 1/(Z-1)$. This is also easily seen from the first derivative of $f(p)$.

$$\displaystyle \begin{aligned} f'(p) &= (1-p)^{Z-2} - p(Z-2)(1-p)^{Z-3} \end{aligned} $$

(3.30)

$$\displaystyle \begin{aligned} & = (1-p)^{Z-3}(1 -p - p(Z-2)) \end{aligned} $$

(3.31)

$$\displaystyle \begin{aligned} & = (1-p)^{Z-3}(1 - (Z-1)p) \end{aligned} $$

(3.32)

Fig. 3.4

A plot $f(p) = p(1-p)^{Z-2}$, which is a term in the cluster number density $n(s,p) = g_s [ p (1-p)^{Z-2} ]^s (1-p)^2$ for the Bethe lattice. We notice that $f(p)$ has a maximum at $p=p:c$, and that the second derivative, $f''(p)$, is zero in this point. A Taylor expansion of $f(p)$ around $p=p:c$ will therefore have a second order term in $(p-p:c)$ as the lowest-order term—to lowest order it is a parabola at $p = p_c$. It is this second order term which determines the exponent $\sigma$, which consequently is independent of Z

which shows that $f'(p_c) = 0$. (We leave it to the reader to show that $f''(p_c) < 0$.) The Taylor expansion of $f(p)$ around $p=p:c$ is then:

$$\displaystyle \begin{aligned} f(p) = f(p_c) + f'(p_c)(p - p_c) + \frac{1}{2} f''(p_c)(p - p_c)^2 + \mathcal{O}((p-p_c)^3) \; , \end{aligned} $$

(3.33)

where we already have found the first order term, $f'(p_c) = 0$. We can therefore write

$$\displaystyle \begin{aligned} f(p) \simeq f(p_c) - \frac{1}{2}f''(p_c)(p - p_c)^2 = A ( 1 - B(p - p_c)^2) \: . \end{aligned} $$

(3.34)

Cluster Number Density

We will now insert this Taylor expansion back into the expression for the cluster number density:

$$\displaystyle \begin{aligned} n(s,p) = g_s [f(p)]^s (1-p)^2 = g_s e^{s \ln f(p)} (1-p)^2\; , \end{aligned} $$

(3.35)

where we now insert $f(p) \simeq A(1 - B(p -p_c)^2)$ to get

$$\displaystyle \begin{aligned} n(s,p) \simeq g_s A^s e^{s \ln (1 - B(p -p_c)^2)} (1-p)^2 \; . \end{aligned} $$

(3.36)

When p is close to $p:c$, $(p-p:c)^2$ is small number, and we can use the first order of the Taylor expansion of $\ln (1-x) \simeq -x + \mathcal {O}(x^2)$, to get

$$\displaystyle \begin{aligned} n(s,p) \simeq g_s A^s e^{-sB(p-p_c)^2}(1-p)^2 \; . \end{aligned} $$

(3.37)

Consequently, for $p = p_c$ we get

$$\displaystyle \begin{aligned} n(s,p_c) = g_s A^s (1-p_c)^2 \; . \end{aligned} $$

(3.38)

Cluster Number Density Expressed in Terms of $n(s,p:c)$

When p is close to $p:c$, we can assume that $(1-p)^2 \simeq (1-p_c)^2$, and we can therefore rewrite the cluster number density in terms of $n(s,p:c)$, giving

$$\displaystyle \begin{aligned} n(s,p) = n(s,p_c) e^{-sB(p - p_c)^2} \; . \end{aligned} $$

(3.39)

We rewrite the exponential in terms of a characteristic cluster size $s_{\xi }$ as

$$\displaystyle \begin{aligned} n(s,p) = n(s,p_c) e^{-s/s_{\xi}} \; , \end{aligned} $$

(3.40)

where the characteristic cluster size $s_{\xi }$ is

$$\displaystyle \begin{aligned} s_{\xi} = B^{-1}(p - p_c )^{-2} \; . \end{aligned} $$

(3.41)

This implies that the characteristic cluster size $s_{\xi }$ diverges as a power-law with exponent $1/\sigma = 2$ as p approaches $p:c$. The general scaling form for the characteristic cluster size $s_{\xi }$ is

$$\displaystyle \begin{aligned} s_{\xi} \propto | p - p_c|^{-1/\sigma} \; , \end{aligned} $$

(3.42)

where the exponent $\sigma$ is universal, which means that is does not depend on lattice details such a Z, but it does depend on lattice dimensionality. It will therefore have a different value for two-dimensional percolation.

Scaling Ansatz for $n(s,p:c)$

We can use our knowledge of the behavior of P and S when p approaches $p:c$ to constrain the scaling behavior of $n(s,p:c)$. We know that we can find S and P from the cluster number density. The average cluster size S as p approaches $p:c$ is

$$\displaystyle \begin{aligned} S = \frac{\varGamma}{p_c - p} \; , \end{aligned} $$

(3.43)

which should diverge when p approaches $p:c$. We rewrite S in terms of $n(s,p)$:

$$\displaystyle \begin{aligned} S = \sum s^2 n(s,p) \; , \end{aligned} $$

(3.44)

which should diverge as p approaches $p:c$. We rewrite the sum as an integral in the limit of $p=p:c$

$$\displaystyle \begin{aligned} S = \int_0^{\infty} s^2 n(s,p_c) ds \; . \end{aligned} $$

(3.45)

This integral must diverge. We can therefore conclude that $n(s,p:c)$ is not an exponential, otherwise the integral then would converge. We therefore assume that the cluster number density has a power-law shape, that is, we introduce the scaling ansatz:

$$\displaystyle \begin{aligned} n(s,p_c) \simeq C s^{-\tau} \; . \end{aligned} $$

(3.46)

This expression is only valid in the limit when $s \gg 1$. We introduce this scaling ansatz into the expression for P:

$$\displaystyle \begin{aligned} \sum_s s n(s,p) = p - P \; , \end{aligned} $$

(3.47)

and approximate the sum with an integral in the limit when $p = p_c$:

$$\displaystyle \begin{aligned} \int_0^{\infty} s n(s,p_c) \mathrm{d} s = \int_0^{\infty} s C s^{-\tau} \mathrm{d} s = p - P \; , \end{aligned} $$

(3.48)

which should converge. We therefore have two conditions

• 1. The integral

  $$\displaystyle \begin{aligned} \int_0^{\infty} s^2 n(s,p_c) ds = \int_0^{\infty} s^2 C s^{-\tau} \mathrm{d} s \end{aligned} $$

  (3.49)

  should diverge, which implies that $\tau -2 \le 1$

• 1. The integral

  $$\displaystyle \begin{aligned} \int_0^{\infty} s n(s,p_c) \mathrm{d} s = \int_0^{\infty} s C s^{-\tau} \mathrm{d} s \end{aligned} $$

  (3.50)

  should converge, which implies that $\tau -1 >1$.

This implies that the exponent $\tau$ has the following bounds:

$$\displaystyle \begin{aligned} 2 < \tau \le 3 \; . \end{aligned} $$

(3.51)

Scaling Behavior of $S(p)$ When p Is Close to $p:c$

We can sum up our arguments so far in the relation

$$\displaystyle \begin{aligned} n(s,p) = n(s,p_c) e^{-B(p-p_c)^2 s} = C s^{-\tau} e^{-B (p - p_c)^2 s} = C s^{-\tau} e^{-s/s_{\xi}} \; . \end{aligned} $$

(3.52)

Let us use this expression to calculate S, for which we know the exact scaling behavior, and then again use this to find the value for $\tau$

$$\displaystyle \begin{aligned} S = C \sum_s s^{2 - \tau} e^{-s / s_{\xi}} \rightarrow C \int_1^{\infty} s^{2 - \tau} e^{-s / s_{\xi}} ds \; . \end{aligned} $$

(3.53)

We now make a rough estimate. This is useful, since it is in the spirit of this book, and it also provides the correct behavior. We assume that

$$\displaystyle \begin{aligned} S = C \int_1^{\infty} s^{2 - \tau} e^{-s / s_{\xi}} ds \sim C \int_1^{s_{\xi}} s^{2-\tau} ds \sim s_{\xi}^{3 - \tau} \; , \end{aligned} $$

(3.54)

where we have used the sign $\sim$ to denote that the expressions have the same scaling behavior. We can do it slightly more elaborately:

$$\displaystyle \begin{aligned} S \simeq C \int_{1}^{\infty} s^{2 - \tau} e^{-s/s_{\xi}} ds \; . \end{aligned} $$

(3.55)

We change variables by introducing, $u = s/s_{\xi }$, which gives

$$\displaystyle \begin{aligned} S \simeq s_{\xi}^{3 - \tau} \int_{1/s_{\xi}}^{\infty} u^{2 - \tau} e^{-u} du \; . \end{aligned} $$

(3.56)

This integral is simply a number, since $1/s_{\xi } \rightarrow 0$, when $p \rightarrow p_c$. The asymptotic scaling behavior in the limit $p \rightarrow p_c$ is therefore

$$\displaystyle \begin{aligned} S \sim s_{\xi}^{3 - \tau} \sim (p - p_c)^{-2(3 - \tau)} \sim (p - p_c)^{-1} \; , \end{aligned} $$

(3.57)

where we have used that

$$\displaystyle \begin{aligned} s_{\xi} \sim (p - p_c)^{-2} \; , \end{aligned} $$

(3.58)

and that

$$\displaystyle \begin{aligned} S \sim (p - p_c)^{-1} \; . \end{aligned} $$

(3.59)

Our direct solution therefore gives that

$$\displaystyle \begin{aligned} \tau = \frac{5}{2} \; . \end{aligned} $$

(3.60)

This relation indeed satisfies the exponent relations we found above, since $2 < 5/2 \le 3$. A plot of the scaling form is shown in Fig. 3.5.

Fig. 3.5

A plot of $n(s,p) = s^{-\tau }\exp (-s(p-p_c)^2)$ as a function of s for various values of p illustrates how the characteristic cluster size $s_{\xi }$ appears as a cut-off in the cluster number density

Preliminary Scaling Theory for Cluster Number Density

We have now developed a preliminary scaling theory for the cluster number density. In the coming chapters, we will demonstrate that similar scaling relations also are valid for percolation in other dimension. We have found that in the vicinity of $p:c$, we do not expect deviations until we reach large s, that is, until we reach a characteristic cluster size $s_{\xi }$ that increases as $p \rightarrow p_c$. The general scaling form for the cluster number density is

$$\displaystyle \begin{aligned} n(s,p) = n(s,p_c) F(\frac{s}{s_{\xi}}) \; ,\ \end{aligned} $$

(3.61)

where

$$\displaystyle \begin{aligned} n(s,p_c) = C s^{-\tau} \; , \end{aligned} $$

(3.62)

and

$$\displaystyle \begin{aligned} s_{\xi} = s_0 | p - p_c|^{-1/\sigma} \; . \end{aligned} $$

(3.63)

In addition, we have the following scaling relations:

$$\displaystyle \begin{aligned} P(p) \sim (p - p_c)^{\beta} \; , \end{aligned} $$

(3.64)

$$\displaystyle \begin{aligned} \xi \sim | p - p_c|^{-\nu} \; , \end{aligned} $$

(3.65)

and

$$\displaystyle \begin{aligned} S \sim | p - p_c|^{-\gamma} \; , \end{aligned} $$

(3.66)

with a possible non-trivial behavior for higher moments of the cluster density.

Exercises

Exercise 3.1 ($P(p)$ for $Z = 4$)

Find $P(p)$ for $Z = 4$ and determine $\beta$ for this value of Z.