Subset Geometry

Abstract

So far, we have studied the geometry of the percolation system. Now, we will gradually address the physics of processes that occur in a percolation system. We have addressed one physics-like property of the system, the density of the spanning cluster, and we found that we could build a theory for the density P as a function of the porosity (occupation probability) p of the system. In order to address other physical properties, we need to have a clear description of the geometry of the percolation system close to the percolation threshold. In this chapter, we will develop a simplified geometric description that will be useful, indeed essential, when we discuss physical process in disordered media. We will introduce various subsets of the spanning cluster—sets that play roles in specific physical processes.

So far, we have studied the geometry of the percolation system. Now, we will gradually address the physics of processes that occur in a percolation system. We have addressed one physics-like property of the system, the density of the spanning cluster, and we found that we could build a theory for the density P as a function of the porosity (occupation probability) p of the system. In order to address other physical properties, we need to have a clear description of the geometry of the percolation system close to the percolation threshold. In this chapter, we will develop a simplified geometric description that will be useful, indeed essential, when we discuss physical process in disordered media. We will introduce various subsets of the spanning cluster—sets that play roles in specific physical processes. We will start by introducing singly connected bonds, the backbone and dangling ends and provide a simplified image of the spanning cluster in terms of the blob model for the percolation system [2, 9, 16, 34].

8.1 Singly Connected Bonds

We will start with an example of a subset of the spanning cluster, the set of singly connected sites (or bonds). This will demonstrate what we mean by a subset and how the subset is connected to a physical problem.

Singly Connected Site A singly connected site is a site with the property that if it is removed, the spanning cluster will no longer be spanning.

We can relate this to a physical property: If we study fluid flow in the spanning cluster, all the fluid has to go through the singly connected sites. These sites are also often referred to as red sites, because if we were studying a set of random resistors, the highest current would have to go through the singly connected bonds, and they would therefore heat up and become “red”. Several examples of subsets of the spanning cluster, including the singly connected bonds, are shown in Fig. 8.1.

Fig. 8.1

Illustration of the spanning cluster, the singly connected bonds (red), the backbone (blue), and the dangling ends (green) for a $256 \times 256$ bond percolation system at $p = p_c$. (Figure from Martin Søreng)

Scaling Hypothesis

We have learned that the spanning cluster may be described by the mass scaling relation $M \propto L^D$, where D is termed the fractal dimension of the spanning cluster. Here, we will make a daring hypothesis, which we will also substantiate: We propose that subsets of the spanning cluster obey similar scaling relations.

For example, we propose that the mass of the singly connected sites ($M:{SC}$) has the scaling form

$$\displaystyle \begin{aligned} M_{SC} \propto L^{D_{SC}} \; , \end{aligned} $$

(8.1)

where we call the dimension $D:{SC}$ the fractal dimension of the singly connected sites. Because the set of singly connected sites is a subset of the spanning cluster, we know that $M_{SC} \le M$. It therefore follows that

$$\displaystyle \begin{aligned} D_{SC} \le D \; . \end{aligned} $$

(8.2)

Based on this simple example, we will generalize the approach to other subsets of the spanning cluster. However, first we will introduce a new concept, a self-avoiding path on the spanning cluster.

8.2 Self-Avoiding Paths on the Cluster

The study of percolation is the study of connectivity, and many of the physical properties that we are interested in depends on various forms of connecting paths on the spanning cluster between two opposite edges. We can address the structure of connected paths between the edges by studying self-avoiding paths (SAPs). A Self-Avoiding Path (SAP) is a set of connected sites that correspond to the sites on the path of a walk on the spanning cluster that does not intersect itself going from one side to the opposite side.

8.2.1 Minimal Path

The shortest path between the two edges is called the shortest SAP between the two edges. (Notice, that there may be more than one path the satisfy this criterion. We chose one of these paths randomly). We call this the minimal path and denote its length $L_{\min }$. The length here refers to the number of sites in the path, which we also call the mass of the path, $M_{\min } = L_{\min }$. We will use mass instead of length in the following to describe the paths.

We assume that mass of the minimal path also scales with the system size according to the scaling form:

$$\displaystyle \begin{aligned} M_{\min} \propto L^{D_{\min}} \; . \end{aligned} $$

(8.3)

Where we have introduced the scaling exponent of the minimal path to be $D_{\min }$.

8.2.2 Maximum and Average Path

Similarly, we call the longest SAP between the two edges the longest path with a mass $M_{\max }$. Again, we assume that the mass has a scaling form, $M_{\max } \propto L^{D_{\max }}$. We notice that $M_{min} \leq M_{max}$. Consequently, a similar relation holds for the exponents $D_{\min } \leq D_{\max }$.

We also introduce the term the average path, meaning the average mass (length) of all possible SAPs going between opposite sides of the system, $\langle M_{SAP} \rangle \propto L^{D_{SAP}}$. The dimension $D:{SAP}$ will lie between the dimensions of the minimal and the maximal path.

8.2.3 Backbone

Intersection of All Self-Avoiding Paths

The notion of SAPs can also be used to address the physical properties of the cluster, such as we saw for the singly connected bonds. The set of singly connected bonds is the set of intersections between all SAPs connecting the two sides. That is, the singly connected bonds is the set of points that any path must go through in order to connect the two sides. From this definition, we notice that the dimension $D_{SC} < D_{\min }$, and as we will see further on, $D_{SC} = 1/\nu$ which is smaller than 1 for two-dimensional systems.

Union of All Self-Avoiding Paths

Another useful set is the union of all SAPs that connect the two edges of the cluster. This set is called the backbone with dimension $D:B$.

Backbone The backbone is the union of all self-avoiding paths on the spanning cluster that connect two opposite edges.

This set has a simple physical interpretation for a random porous material, since it corresponds to the sites that are accessible to fluid flow if a pressure is applied across the material. The remaining sites are called dangling ends. The backbone are all the sites that have at least two different paths leading into them, one path from each side of the cluster. The remaining sites only have one (self-avoiding) path leading into them, and we call this set of sites the dangling ends. The spanning cluster consists of the backbone plus the dangling ends, as illustrated in Fig. 8.2. The dangling ends are therefore pieces of the cluster that can be cut away by the removal of a single bond.

Fig. 8.2

Illustration of the spanning cluster consisting of the backbone (red) and the dangling ends (blue) for a $512 \times 512$ site percolation system for (a) $p = 0.58$, (b) $p = 0.59$, and (c) $p = 0.61$

We have arrived at the following hierarchy of exponents describing various subsets of paths through the cluster:

$$\displaystyle \begin{aligned} D_{SC} \leq D_{\min} \leq D_{SAP} \leq D_{\max} \leq D_B \leq D \leq d \; , {} \end{aligned} $$

(8.4)

8.2.4 Scaling of the Dangling Ends

Generally, we will find that the dimension of the backbone, $D:B$, is smaller than the dimension of the spanning cluster. For example, in two dimensions, we find that $D_B \simeq 1.6$, whereas $D \simeq 1.89$. This has implications for the relative size of the backbone and the dangling ends.

The spanning cluster consists of the backbone and the dangling ends. Therefore, the mass of the spanning cluster, M, must equal the sum of the masses of the backbone and the dangling ends $M = M_B + M_{DE}$. Since we know that $M \propto L^{D}$ and $M_B \propto L^{D_B}$, we find that

$$\displaystyle \begin{aligned} M_{DE} = M - M_B = M_0 L^{D} - M_{0,B} L^{D_B} \; , \end{aligned} $$

(8.5)

where $M:0$ and $M:{0,B}$ are constant prefactors. To see what happens when $L \rightarrow \infty$, we divide by M:

$$\displaystyle \begin{aligned} \frac{M_{DE}}{M} = 1 - \frac{M_{0,B}L^{D_B}}{M_0 L^{D}} = 1 - c L^{D_B - D} \; , \end{aligned} $$

(8.6)

Since $D_B \leq D$, we see that the fraction $M:{DE}/M$ goes to a constant (one) as L approaches infinity. Consequently, we have found that $M_{DE} \propto M \propto L^{D}$. This also implies that as the system size goes to infinity most of the mass is in the dangling ends. This means that the backbone occupies a smaller and smaller portion of the total mass of the system as the system size increases.

8.2.5 Argument for the Scaling of Subsets

We can provide a better argument for why the various subsets should scale with the system size L to various exponents. We notice that the following relation between the masses must be true:

$$\displaystyle \begin{aligned} L^1 \leq M_{\min} \leq M_{SAP} \leq M_{\max} \leq M_{BB} \leq M \leq L^d \; , \end{aligned} $$

(8.7)

where the first inequality $L^1 \leq M_{\min }$ follows because even the minimum path must be at least of length L to go from one side to the opposite side.

Now, if this is to be true for all values of L, it can be argued that because all the masses are between two scaling relations, $L^1$ and $L^d$, also the scaling of the intermediate masses, $M:x$, must be power-laws with some power-law exponents, $M_x \propto L^{D_x}$, with the hierarchy of exponents given in (8.4).

8.2.6 Blob Model for the Spanning Cluster

Let us now try to formulate our geometric description of the spanning cluster into a model of the spanning cluster [36]. We have found that the spanning cluster can be subdivided first into two parts: the backbone and the dangling ends. The backbone may again be divided into two parts: a set of blobs where the are several parallel paths and a set of sites, the singly connected sites, that connect the blobs to each other and the blobs to the dangling ends. Thus, we have ended up with a model with three components:

• the dangling ends,

• a set of blobs where there are several parallel paths

• the singly connected points, connecting the blobs to each other and the blobs to the dangling ends.

Each of the blobs and the dangling ends will again have a similar substructure of dangling ends, blobs with parallel paths, and singly connected bonds as illustrated in Fig. 8.3. This cartoon image of the clusters provides very useful intuition about the geometrical structure of percolation clusters, which we will use when we address the physics of disordered systems in the next chapters.

Fig. 8.3

Illustration of the hierarchical blob-model for the percolation cluster showing the backbone (bold), singly connected bonds (red) and blobs (blue)

8.2.7 Mass-Scaling Exponents for Subsets of the Spanning Clusters

The exponents can be calculated either by numerical simulations, where the masses of the various subsets are measured as a function of system size at $p=p:c$, or by the renormalization group method. Numerical results based on computer simulations using the code provided in this book are listed in Table 8.1. You can find up-to-date results for exponents in the percolation system at the Wikipedia page: https://en.wikipedia.org/wiki/Percolation_critical_exponents.

Table 8.1 A list of known exponent for the various subset types in two dimensions

8.3 Renormalization Calculation

We will now use the renormalization group approach to address the scaling exponent for various subsets of the spanning cluster at $p = p_c$. For this, we will here use the renormalization procedure for bond percolation on a square lattice in two dimensions following Hong and Stanley [17], where we have found that the renormalization procedure produces the exact result for the percolation threshold, $p_c = p^{\ast } = 1/2$, which is a fixpoint of the mapping.

Our strategy will be to assume that all the bonds have a mass $M = 1$ in the original lattice, and then find the mass $M'$ in the renormalized lattice, when the length has been rescaled by b. For a property that displays a self-similar scaling, we will expect that

$$\displaystyle \begin{aligned} M' \propto b^{D_x} M \; , \end{aligned} $$

(8.8)

where $D:x$ denotes the dimension for the particular subset we are looking at. We can use this to determine the fractal exponent $D:x$ from

$$\displaystyle \begin{aligned} D_x = \frac{\ln M'/M}{\ln b} \; . \end{aligned} $$

(8.9)

We will do this by calculating the average value of the mass of the H-cell, by taking the mass of the subset we are interested in for each configuration, $M:x(c)$, and multiplying it by the probability of that configuration, summing over all configurations:

$$\displaystyle \begin{aligned} \langle M \rangle = \sum_c M_x(c) P(c) \; . \end{aligned} $$

(8.10)

We have now calculated the average mass in the original 2 by 2 lattice, and this should correspond to the average renormalized mass, $\langle M' \rangle = p' M'$, which is the mass of the renormalized bond, $M'$ multiplied with the probability for that bond to be present $p'$. That is, we will find $M'$ from:

$$\displaystyle \begin{aligned} p' M' = \sum_c M(c) P(c) \; , \end{aligned} $$

(8.11)

We will study our system at the nontrivial fixpoint $p = p^{\ast } = 1/2 = p_c$. The spanning configurations c for bond renormalization in two dimensions, are shown together with their probabilities and the masses of various subsets in Table 8.2.

Table 8.2 Numerical exponents for the exponent describing various subsets of the spanning cluster defined using the set of Self-Avoiding Walks going from one side to the opposite side of the cluster. The last line shows the exponents found from numerical simulations in a two-dimensional system

This use of the renormalization group method to estimate the exponents demonstrates the power of the renormalization arguments. Similar arguments will be used to address other properties of the percolation system.

8.4 Deterministic Fractal Models

We have found that we can calculate the behavior of infinite-dimensional and one-dimensional systems exactly. However, for finite dimensions such as for $d=2$ or $d=3$ we must rely on numerical simulations and renormalization group arguments to determine the exponents and the behavior of the system. However, in order to learn about physical properties in systems with scaling behavior, we may be able to construct simpler models that contain many of the important features of the percolation cluster. For example, we may be able to introduce deterministic, iterative fractal structures that reproduce many of the important properties of the percolation cluster at $p = p_c$, but that are deterministic systems. The idea is that we can use such a system to study other properties of the physics on fractal structures.

Mandelbrot-Given Curve

An example of an iterative fractal structure that has many of the important features of the percolation clusters at $p = p_c$ is the Mandelbrot-Given curve. The curve is generated by the iterative procedure described in Fig. 8.4. Through each generation, the length is rescaled by a factor $b = 3$, and the mass is rescaled by a factor 8. That is, for generation l, the mass is $m(l) = 8^l$, and the linear size of the cluster is $L(l) = 3^l$. If we assume a scaling on the form $m = L^D$, we find that

$$\displaystyle \begin{aligned} D = \frac{\ln 8}{\ln 3} \simeq 1.89 \; . \end{aligned} $$

(8.12)

This is surprisingly similar to the fractal dimension of the percolation cluster. We can also look at other dimensions, such as for the singly connected bonds, the minimum path, the maximum path and the backbone.

Fig. 8.4

Illustration of first three generations of the Mandelbrot-Given curve. The length is scaled by a factor $b=3$ for each iteration, and the mass of the whole structure is increased by a factor of 8. The fractal dimension is therefore $D = \ln 8 /\ln 3 \simeq 1.89$

Single Connected Bonds

Let us first address the singly connected bonds. In the zero’th generation, the system is simply a single bond, and the length of the singly connected bonds, $L:{SC}$ is 1. In the first generation, there are two bonds that are singly connecting, and in the second generation there are four bonds that are singly connecting. The general relation is that

$$\displaystyle \begin{aligned} L_{SC} = 2^l \; , \end{aligned} $$

(8.13)

where l is the generation of the structure. The dimension, $D:{SC}$, of the singly connected bonds is therefore $D_{SC} = \ln 2 / \ln 3 \simeq 0.63$, which should be compared with the exact value $D_{SC} = 3/4$ for two-dimensional percolation.

Minimum Path

The minimum path will for all generations be the path going straight through the structure, and the length of the minimal path will therefore be equal to the length of the structure. The scaling dimension $D:{min}$ is therefore $D_{min} = 1$.

Maximum Path

The maximum path increases by a factor 5 for each iteration. The dimension of the maximum path is therefore $D_{max} = \ln 5 / \ln 3 \simeq 1.465$.

Backbone

We can similarly find that the mass of the backbone increases by a factor 6 for each iteration, and the dimension of the backbone is therefore $D_{B} = \ln 6 / \ln 3 \simeq 1.631$.

Model System

This deterministic iterative fractal can be used to perform quick calculations of various properties on a fractal system, and may also serve as a useful hierarchical lattice on which to perform simulations when we are studying processes occurring on a fractal structure.

8.5 Lacunarity

The fractal dimension describes the scaling properties of structures such as the percolation cluster at $p=p:c$. However, structures that have the same fractal dimension, may have a very different appearance. As an example, let us study several variations of the Sierpinski gasket introduced in Sect. 5.3. As illustrated in Fig. 8.5, we can construct several rules for the iterative generation of the fractal that all result in the same fractal dimension, but have different visual appearance. The fractal dimension $D = \ln 3 / \ln 2$ for both of the examples in Fig. 8.5, but by increasing the number of triangles that are used in each generation, the structures become more homogeneous. How can we quantify this difference?

Fig. 8.5

Two versions of the Siepinski gasket. In version 1, the next generation is made from 3 of the structures from the last generation, and the spatial rescaling is by a factor $b=3$. In version 2, the next generation is made from 9 of the structures from the last generation, and the spatial rescaling is by a factor $b=6$. The resulting fractal dimension is $D_2 = \ln 9 / \ln 4 = \ln 3^2 / \ln 2^2 = \ln 3/\ln 2 = D_1$. The two structures therefore have the same fractal dimension. However, version 1 have large fluctuations that version 2

Distribution of Mass

In order to quantify this difference, Mandelbrot invented the concept of lacunarity . We measure lacunarity from the distribution of mass-sizes. We can characterize and measure the fractal dimension of a fractal structure using box-counting, as explained in Sect. 5.3. The structure, such as the percolation cluster, is divided into boxes of size $\ell$. In each box, i, there will be a mass $m_i(\ell )$. The fractal dimension is found by calculating the average mass per box of size $\ell$:

$$\displaystyle \begin{aligned} \langle m_i(\ell) \rangle_i = A \ell^D \; . \end{aligned} $$

(8.14)

However, there will variations in the masses $m(\ell )$ in the boxes, characterized by a distribution $P(m,\ell )$, which gives the probability for mass m in a box of size $\ell$. We can characterize this distribution by its moments:

$$\displaystyle \begin{aligned} \langle m^k (\ell) \rangle = A_k \ell^{kD} \; , \end{aligned} $$

(8.15)

where this particular scaling form implies that the structure is unifractal: the scaling exponents for all the moments are linearly related.

Unifractal Scaling

For a unifractal structure, we expect the distribution of masses to have the scaling form

$$\displaystyle \begin{aligned} P(m,\ell) = \ell^x f(\frac{m}{\ell^D}) \; , \end{aligned} $$

(8.16)

where the scaling exponent x is yet undetermined. In this case, the moments can be found by integration over the probability density

$$\displaystyle \begin{aligned} \langle m^k \rangle & = \int P(m,\ell) m^k \, \mathrm{d} m \end{aligned} $$

(8.17)

$$\displaystyle \begin{aligned} & = \int m^k \ell^x f(\frac{m}{\ell^D}) \, \mathrm{d} m \end{aligned} $$

(8.18)

$$\displaystyle \begin{aligned} & = \ell^{(kD + x + D} \int (\frac{m}{\ell^D})^k f(\frac{m}{\ell^D}) \, \mathrm{d} (\frac{m}{\ell^D}) \end{aligned} $$

(8.19)

$$\displaystyle \begin{aligned} & = \ell^{D(k+1) + x} \int x^k f(x) \, \mathrm{d} x \end{aligned} $$

(8.20)

We can determine the unknown scaling exponent x from the scaling of the zero’th moment, that is, from the normalization of the probability density: $\langle m^0 \rangle = 1$ implies that $D(0+1)+x=0$, and therefore, that $x = -D$. The scaling ansatz for the distribution of masses is therefore

$$\displaystyle \begin{aligned} P(m,\ell) = \ell^{-D} f(\frac{m}{\ell^D}) \; . \end{aligned} $$

(8.21)

And we found that the moments can be written as

$$\displaystyle \begin{aligned} \langle m^k \rangle = \ell^{D(k+1) -D} \int x^k f(x) dx = A_k \ell^{kD} \; , \end{aligned} $$

(8.22)

as we assumed above. Consequently, the distribution of masses is characterized by the distribution $P(m,\ell )$, which in turn is described by the fractal dimension, D, and the scaling function $f(u)$, which gives the shape of the distribution.

Properties of the Distribution of Masses

The distribution of masses can be broad, which would correspond to “large holes”, or narrow, which would correspond to a more uniform distribution of mass. The width of the distribution can be characterized by the mean-square deviation of the mass from the average mass:

$$\displaystyle \begin{aligned} \varDelta = \frac{\langle m^2 \rangle - \langle m \rangle^2}{\langle m \rangle^2} = \frac{A_2 - A_1^2}{A_1^2} \; . \end{aligned} $$

(8.23)

This number describes another part of the mass distribution relation than the scaling relation, and can be used to characterize fractal set. For the percolation problem, this number is assumed to be universal, independent of lattice type, but dependent on the embedding dimensionality of the system.

Exercises

Exercise 8.1 (Singly Connected Bonds)

Use the example programs from the text to find the singly connected bonds.

• (a) Run the programs to visualize the singly connected bonds. Can you understand how this algorithms finds the singly connected bonds? Why are some of the bonds of a different color?

• (b) Find the mass, $M:{SC}$, of the singly connected bonds as a function of system size L for $p=p:c$ and use this to estimate the exponent $D:{SC}$: $M_{SC} \propto L^{D_{SC}}$.

• (c) Can you find the behavior of $P_{SC} = M_{SC}/L^d$ as a function of $p - p_c$?

Exercise 8.2 (Left/Right-Turning Walker)

We have provided a subroutine and an example program that implements the left/right-turning walker algorithm. The algorithm works on a given clusters. From one end of the cluster, two walkers are started. The walkers can only walk according to the connectivity rules on the lattice. That is, for a nearest-neighbor lattice, they can only walk to their nearest neighbors. The left-turning walker always tries to turn left from its previous direction. If this site is empty, it tries the next-best site, which is to continue straight ahead. If that is empty, it tries to move right, and if that is empty, it moves back along the direction it came. The right-turning walker follows a similar rule, but prefers to turn right in each step. The first walker to reach the other end of the cluster stops, and the other walker stops when it reaches this site.

The path of the two walkers is illustrated in the Fig. 8.6. The sites that are visited by both walkers constitute the singly connected bonds. The union of the two walks constitutes what is called the external perimeter (Hull) of the cluster.

Fig. 8.6

Illustrations of the left-right turning walker

• (a) Use the following programs to generate and illustrate of the singly connected bonds for a $100 \times 100$ system. Check that the illustrated bonds correspond to the singly connected bonds.

• (b) Measure the dimension $D:{SC}$.

• (c) Modify the programs to find the external perimeter (Hull) of a spanning cluster in a $100 \times 100$ system.

• (d) Measure the dimension $D:P$ of the perimeter.

• (e) (Advanced) Develop a theory for the behavior of $P:H(p,L)$, the probability for a site to belong to the Hull as a function of p and L for $p>p_c$.

• (f) (Advanced) Measure the behavior of $P:H(p,L)$ as a function of p for $L = 512 \times 512$.