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Percolation

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DOI: 10.5047/forma.2015.s004

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Review Forma, 30, S17–S23, 2015

Percolation
Sasuke Miyazima

Department of Natural Sciences, Chubu University, Matsumoto-cho 1200, Kasugai, Aichi 487-8501, Japan
E-mail address: miyazima@isc.chubu.ac.jp

(Received September 30, 2014; Accepted January 19, 2015)

Percolation model is one of the most foundamental models holding important concept such as phase transition,
growth phenomena, universality, and also it provides clues for studies of transport coefficients of porous media,
forest fire, epidemics in an orchard, networks and so on. There are mainly two types of percolation models,
discrete percolation model and continuum percolation model. The former is based on a lattice structure, where
some exact solutions are found, while the latter on continuum space, where more experimental investigations are
expected.
Key words: Discrete Percolation Model, Continuum Percolation Model, Critical Point, Critical Exponent, Fractal

1. Introduction the directed percolation model (Hinrichsen, 2000), kinetic

Elements such as particles, bonds etc. are placed onto growth model (Gould et al., 2005), forest fire mode (Hen-
lattice sites (or bonds) in a space one by one. If these el- ley, 1993), and invasion percolation model (Wilkinson and
ements occupy the nearest neighboring positions, then they Willemsen, 1983). In addition, a quantum percolation
form clusters. The cluster becomes larger and larger with model (de Gennes et al., 1959) is also investigated, but here
increase of the elements, and at some quantity of elements, we don’t discuss these problems. Rather we stress on some
the cluster acquires a size extending from an end to oppo- percolation models with exact solutions. In the 2D the crit-
site end in a finite space or becomes infinitely large in an ical points and the critical exponent are calculated for some
infinite space. We call this phenomenon percolation (Es- limited lattices. We are going to discuss some percolation
sam, 1980; Grimmett, 1989; Stauffer and Aharony, 1994; models with exact critical values in higher dimensions.
Sahimi, 1994; Bunde and Havlin, 1996; Hunt, 2013) and In the above discussion we suppose that the elements are
this model is called a the discrete percolation model. If the placed on the lattice point of regular lattices or the elements
element has electric conductivity, the whole media becomes are replaced by other elements. But there are other fami-
conductive at the instance of percolation. This phenomenon lies of percolation models where the elements are placed at
shows critical behavior as shown in the text. Most impor- arbitrary points in the space. We call this category of per-
tant indications are critical point ( pc ) and critical exponent colation models continuum percolation model (Meester,
(ν). The former depends on the details of lattice when the 1996). The most important point of this continuum perco-
element occupies only lattice point or only bond between lation model is the critical exponents which are generally
the nearest neighboring two points. The latter quantity does different from those in the discrete model, which means an
not depend on the species of lattice, but does on the space existence of a different class of universality from that in the
dimension (universality). discrete percolation model.
Most of investigations of percolation model are per- In nature there are many media for which the continuum
formed by computer simulation, but one of most simple percolation model is adequate such as rocks, porous media,
and powerful theoretical approaches may be the real space farm field, flow in vessel and so on (Sahimi, 1994; Hunt,
renormalization group theory which provided the critical 2013). Investigation of the continuum percolation model
points and the critical exponents, but is limited to 2 dimen- has many difficulties for the following reasons. (i) The sim-
sion (Reynolds et al., 1978). When we look at the pattern of ulation method does not work well, mainly because of dif-
percolation at the critical point, it is easy to see the fractal ficulty in calculation of area or volume of overlapping ele-
structure which provides the power law behaviors in sev- ments. (ii) Also, natural samples are limited in variations
eral quantities such as probability that a site belongs to fi- of concentration of elements. (iii) Few experimental inves-
nite size cluster, percolation probability, average cluster size tigations are given in the text.
and so on. Recently the network problems are studied very In the next section the discrete percolation model and
actively, but these problems are profoundly related with per- some new percolation models with the exact value of char-
colation (Barabasi and Alber, 1999). acteristic quantity will be discussed. In Section 3, we will
There are many modified percolation models such as discuss the continuum percolation model.

Copyright

c Society for Science on Form, Japan.

doi:10.5047/forma.2015.s004

S17
S18 S. Miyazima

Table 1. The critical point of several lattices in 2D. ν is the ordination number (Sykes and Essam, 1964; Jacobsen, 2014).

Lattice ν pc of bond model pc of site model

Honeycomb 3 1 − 2 sin(π/18) ∼
= 0.6527∗ 0.6970
Square 4 1/2∗ 0.5927
Kagome 4 0.5244 0.6527∗
Triangular 6 2 sin(π/18) ∼
= 0.3473∗ 0.5∗
∗ Means the exact critical point.

Table 2. The critical point of several lattices in the 3D. ν is the ordination number (Lorenz and Ziff, 1998a, b; Skvor and Nezbeda, 2009; Wang et al.,
2013).

Lattice ν pc of bond model Pc of site model

Simple cubic 6 0.2488 0.3116
Body centered cubic 8 0.1803 0.2465
Face centered cubic 12 0.1202 0.1993

2. Discrete Percolation Model 2.1 The critical point of bond and site percolation
In nature, at first there is a space, and when the space models
is completely filled by the elements, we call it as pure As explained above, in initial state all bonds on a regular
materials. If the element is a conducting lot, disk, or sphere, lattice are closed bonds. If ν N p/2 bonds among all bonds
a cluster may appear and grow in the space as the number are replaced by open bonds, where N , ν, p are the total
of elements increases. If the cluster spreads from one end vertex number, the ordination number of the lattice and
to other end in the space, we call it percolation. When the the ratio of open bond to all bonds, respectively. A few
element added is conductive, the bulk material in the space smaller clusters appear and the size of cluster increases
becomes conductive and the conductance increases with the with p. As a regular 2D lattice there are the square, the
number of elements. To the contrary, we can imagine that triangular, the honeycomb and the Kagome lattice, while
the initial state is conductive and insulating elements are there are the simple cubic, the body-centered cubic and
added. In this case, as the insulator increases in number, the the face centered cubic lattice in 3D. At some value of p,
conductance decreases and it vanishes at the critical point. the cluster becomes end-to-end cluster, where the cluster
Here, the behavior how to decrease is also interesting. This reaches both ends (from top to bottom, or from right to
is a kind of critical phenomena like that in ferromagnetism, left). Or if the lattice is an infinitely large one, the cluster
where a typical model is the Ising Model (Bhattacharjee and becomes infinitely large.
Khare, 1995). The threshold of the percolation model was initially cal-
Percolation phenomena were studied in polymer science culated by simulation. The threshold value depends on the
at first by Flory (1941), where rigid polymer cluster grows species of lattice and its dimension as shown in Tables 1 and
in liquid. In 1956, Broatbent and Hammersley introduced 2.
the mathematical concept of percolation discussing fluid In 2D, we have several exact values, one of those will be
flow in porous medium (Broatbent and Hammersley, 1957). explained in the following subsection. As seen in Tables 1
They considered a regular lattice structure with random ar- and 2, if the ordination number ν increases, the critical point
rangement of open tube or closed tube at the bond of lattice. decreases both in the discrete and the continuum percolation
The fluid can flow in the open tube at the bond, then we call models. On the other hand, the critical exponent ν of the
it open bond simply. On the other hand, the fluid cannot correlation length is 4/3 for 2D and 0.9 for 3D, which are
flow at the closed bond. Hereafter, we call this model as independent of lattice species and dependent only on the
the bond percolation model. space dimensions.
Instead of open or closed bonds in a regular lattice, we 2.2 Real space renormalization group theory
can introduce a plug at the vertex in the lattice with all open The renormalization group theory was originally pro-
bonds. When we image formation of an infinitely large gressed in the elementary particle physics. The first appli-
cluster of polymer molecules, each molecule, which we call cation to the solid state physics was done by Wilson (1971),
simply a particle, can occupy a vertex site. If two particles Amit (1984).
occupy the nearest neighbor vertices, these two particles are In 1975, Harris et al. applied this method to percolation
combined. When more particles are added at vertices, larger in the Fourier space (Harris, 1974). He discussed the same
ensemble of particles may appear. We call this a cluster and problem in the real space (Harris et al., 1975). Here we
this model the site percolation model. These site and bond introduce the latter method because of its intuitive property.
percolation models have become the standard models. Let us consider the site percolation model of the square
 Percolation S19

Fig. 1. 2 × 2 sites (dots) of the square lattice is renormalized into a

renormalized site (X).

Fig. 3. The solid line shows the triangle lattice and the dashed the
honeycomb lattice.

Fig. 2. The solid line shows the real square lattice and the dashed line the
dual one.

lattice, and we choose 2 × 2 sites (small black circles) and Fig. 4. The solid line shows the real Kagome lattice and the dashed line
its renormalized site (denoted by X) as shown in Fig. 1. the dual, which is the diced lattice.
The renormalized site is occupied by renormalized particle
if more than one site are occupied. On the other hand,
if less than two sites are occupied, the renormalized site In the same way, we can consider the bond problem as
is unoccupied. In the case that two sites are occupied, is done by Stauffer and Aharony (1994). We adopt 2 × 2
we share two cases between occupation and un-occupation bonds as shown in Fig. 1. We consider only the horizontal
by renormalization (Stauffer and Aharony, 1994). p  is renomalized bond, then we obtain
renormalized concentration of particles is expressed as
p = p4 + 4 p3 q + 2 p2 q 2 , p = p5 + 5 p4 q + 8 p3 q 2 + 2 p2 q 3 .

where q = 1 − p. At the threshold point we can expect The details of each term in the above equation are given in
p  = p = pc . Therefore, we obtain Stauffer and Aharony (1994). From this equation, we obtain
pc = 0.6180,
pc = 1/2 and ν = 1.428, where b = 2.
which is very near to the value pc = 0.592746.
For p ∼= pc , the correlation length is expressed as ξ ≈ The threshold value is the same as the exact one and the
( p − pc )−ν . If the renomalization process is applied once, correlation exponent is also very near to the exact one.
the bond length is renormalized by the factor 2 and the There exist several trials to extend 2D real space renor-
correlation length is also renormalized as ξ = bξ  ≈ b( p  − malization group theory to the 3D, however, the results were
pc )−ν ≈ ( p − pc )−ν , where b = 2. Therefore not good enough and a better technique is expected. Monte
log b log b Carlo renormalization method is along this direction.
ν=  = . 2.3 An exact method
p − pc d p
lim p→ pc log log 2.3.1 Dual transformation An argument was given
p − pc dp
by Krammers and Wannier to obtain the critical temperature
If we use the critical value pc = 0.6180, we obtain ν = of the Ising model (Kramers and Wannier, 1941) and Harris
1.63, which is very near to the exact value ν = 4/3. applied it to the percolation model (Harris, 1960). The
S20 S. Miyazima

Fig. 5. The unit cell of a 4D hyper-cubic lattice is shown by the solid

lines. The big dot is a common center of two squares. One of them
is the square (0, 1, 1, 0) − (0, 1, 1, 1) − (1, 1, 1, 1) − (1, 1, 1, 0) in the
original lattice. The other one is denoted by dotted lines in the dual
lattice. These two squares are orthogonal each other.
Fig. 6. 2D Swiss-cheese model. Many holes are punched out on aluminum
foil. This picture shows a situation a little before beginning of percola-
partition function of Ising model with N spins is tion from the left side to the right, where the conductivity from the top
to the bottom vanishes.

    J µi µ j 
2J
Z = exp
kT {µ} i, j
kT
    the honeycomb lattice (Fig. 3). The diced lattice and the
J J µi µ j Kagome lattice are mutually dual (Fig. 4).
= cosh N 1 + tanh .
kT {µ} i, j kT This dual relation was used to percolation model in the
square lattice which is the self-dual as seen in Fig. 2. Ini-
Here, if we make a bond of the dual lattice correspond tially we assume that the real lattice is fully occupied with
to a term “tanh{J µi µ j /kT }” and no bond to “1” in the bonds and when we remove a bond randomly from the real
above equation and sum over all the spin arrangement (µ = square lattice, say,
±1), we can easily understand that only closed circuits
made of the bonds contribute to the partition function. This (i, j) − (i, j + 1), or (i, j) − (i + 1, j)
closed circuit means an important equivalence between the
spin arrangement of the original lattice so that a new spin then immediately we place a dual bond
arrangement of σ = −1 corresponds inside the closed
(i + 1/2, j + 1/2) − (i − 1/2, j + 1/2) or
circuit and that of σ = +1 to outside of the closed circuit,
where these new spins are located at the center of each (i + 1/2, j + 1/2) − (i + 1/2, j − 1/2),
square of the original lattice. We call a set of this new
site for new spin a dual lattice which is discussed below. respectively. Therefore, the total number of bonds in both
Furthermore we put the real and the dual square lattices are 2N . If we introduce
p and p  as the concentrations of the real and the dual
J J lattices, respectively, we have
e− kT  = tanh ,
kT
p + p  = 1.
then we can rewrite the above partition function as



 Here, the removed and replaced bonds are orthogonal each
2J 2J N /2 2J
Z = sinh Z . other and the replaced bond is determined without ambigu-
kT kT kT ity. If the percolation model of the square has a phase tran-
If the original lattice is the square lattice, then the dual sition at a concentration value, the other dual lattice must
lattice is also the square lattice. Therefore, if they have have the 
similar transition at the same value, therefore pc
critical temperatures, they must be the same, and we have and p c should be the same. Therefore,
2J/kT = 0.2441. 1
As an example of the percolation model, we treat the pc = pc = .
2
square lattice. At the center of each cell, we assume another
site (we call it as a dual site (or vertex)). Combining all Further mathematically exact treatment was given by
dual sites with dual bonds, we find another square lattice Kesten (1980).
(dual lattice) as shown in Fig. 2. In this case, we obtain the 2.3.2 Application of dual transformation to 3D
same square lattice as a dual lattice (self dual). Applying space In this section we will apply the dual transforma-
the same technique to the triangular lattice, we obtain the tion to cubic lattice in 3D. We choose dual site at the center
honeycomb lattice. And we obtain triangular lattice from of each cubic cell in the real 3D space. These dual sites form
 Percolation S21

Table 3. The critical exponent of several transport coefficients of the continuum percolation models. S.C. and I.S.C. are Swiss-cheese model and Inverse
Swiss-cheese model, respectively. The figures with asterisk are different from those of the discrete percolation model (Feng et al., 1987).

2D S.C. 2D I.S. C 3D S.C. 3D I.S. C

Conductivity 1.3 1.3 2.4∗ 1.9
Elasticity 5.1∗ 1.3 4.4∗ 2.4∗
Permeability 5.1∗ 1.3 6.2∗ 4.2∗

the same cubic lattice in the dual space. In this lattice a cor- is located at (i + 12 , j + 12 , k, l) and the corresponding square
responding element to a bond in the real space is a square in the dual 4D hyper-cubic lattice is (i + 12 , j + 12 , k + 12 , l +
(or parquet) in the dual lattice. First we assume that the 1
) − (i + 12 , j + 12 , k + 12 , l − 12 ) − (i + 12 , j + 12 , k −
2
dual cubic lattice is occupied by parquets. When we choose 1
, l + 12 ) − (i + 12 , j + 12 , k − 12 , l − 12 ). If we choose
a bond in the real cubic lattice, say, (i, j, k) − (i + 1, j, k), 2
a parquet as an example, i.e. (0, 1, 1, 0) − (0, 1, 1, 1) −
the center of this bond is located at (i + 12 , j, k). The corre- (1, 1, 1, 1)−(1, 1, 1, 0), is in the x −ξ plane, where ξ is the
sponding parquet in the dual space (i + 12 , j + 12 , k+ 12 )−(i + fourth axis, then the corresponding parquet in the dual lat-
1
2
, j − 12 , k + 12 )−(i + 12 , j − 12 , k − 12 )−(i + 12 , j + 12 , k − 12 ), tice, ( 12 , 12 , 32 , 12 )−( 12 , 32 , 32 , 12 )−( 12 , 32 , − 12 , 12 )−( 12 , 12 , 12 , 12 ),
is deleted. This process will be repeated untill an infinite is in the y − z plane. These two squares are orthogonal each
chain is accomplished in the real space. Then, if we inject other and the square in the dual lattice is definitely deter-
the water from the top to bottom along the infinite chain in mined.
real space, then we find the water flow in the dual lattice, Here, the dual lattice is a 4D hyper-cubic which is equiv-
although at initial there is no such infinite chain of hole, alent to the original lattice. Therefore, if the latter has a crit-
which means disappearance of infinite surface. The infinite ical value, then the former also has the critical point at the
surface prohibited at initial the flow from top to bottom in same concentration as in the original lattice. As the same
the dual space. manner as before, we assume a dual 4D hyper-cubic lattice
At the same time, the water flow starts from the top to filled with squares. If we delete a square in the dual lattice,
the bottom in the original cubic lattice, the tube for water in we place a corresponding square onto the original lattice.
the dual cubic lattice is formed along the infinite chain of At critical concentration of square of the original lattice, we
bonds in the original lattice. Therefore, the threshold value can expect an infinite surface. At the same time, an infinite
in the cubic lattice with distribution of square is 0.7512 surface in the orthogonal direction disappears in the dual
· · ·. Unfortunately we don’t know the exact value of the 4D hyper-cubic lattice.
threshold value in any 3D lattices. 0.2488 . . . is obtained Since the original lattice and the dual lattice are equiva-
from the numeric method, therefore 0.7512 . . . is also not lent, they should have the same threshold value at Pc = 0.5.
exact one. This result is easily extended to all even 2nD space, where
In the case of site percolation problem, if the nearest we distribute a unit with half dimension n. For example, in
neighboring two sites are occupied, we assume that these the case of 6D space, the unit is 3D, i.e. a cube (Miyazima,
two sites are combined forming a bond. In a similar man- 2005b).
ner, if four sites at the four corners of a parquet are occu-
pied, let us assume they form plane which prohibits water 3. Continuum Percolation Model
flow. As particles are distributed onto the cubic lattice, ini- The percolation model is divided into two types, one is
tially we find single particles, and then two-particle clus- the discrete percolation model discussed in the previous
ters, and so on. They grow with particle concentration. At paragraph. The other is the continuum percolation model
pc1 = 0.2173 . . ., we find an infinite chain. Furthermore as where components are placed on arbitrary position in the
the concentration of particles increases, we will find many space. In nature, however, most of cases, such as rocks with
parquets. At pc2 = 1 − 0.217 . . ., ensemble of parquets conducting parts dispersed in insulating part, or vice versa,
form an infinitely wide film in the cubic lattice. Finally we should be analyzed by adopting the continuum percolation
can say that there are two phase transitions in the cubic lat- model, where the components occupy arbitrary positions in
tice, if we permit formation a bond by two particles and a the 3D space.
parquet by four particles (Miyazima, 2005a). Even if we adopt a discrete percolation model with very
2.3.3 Application of dual transformation to 4D fine lattice, we cannot obtain critical exponents which are
space—An exact threshold value in 4D space— In the obtained by experiments and observations. One of the sim-
similar manner, we consider a dual lattice of a 4D hyper- plest models with this direction is Swiss-cheese model,
cubic lattice. It is easy to see that the dual lattice of a 4- where in the 2D case we open many holes at arbitrary
dimension hyper-cubic lattice is also a 4D hyper-cubic lat- positions in a conducting thin film (see Fig. 6). A mod-
tice. We choose an arbitrary square in the original hyper- ification of the Swiss-cheese model is an inverse Swiss-
cubic lattice with lattice sites expressed as (i, j, k, l) − (i + cheese model, where many conductive circles (or spheres)
1, j, k, l) − (i, j + 1, k, l) − (i + 1, j + 1, k, l), for example, are placed on insulating plate (or into bulk). Mostly elec-
and also other 5 different squares with different directions, tricity, elasticity and permeability are discussed by observa-
where i, j, k, l, are integer. Then, the center of this square tion of rocks, and theoretical and experimental studies.
S22 S. Miyazima

Fig. 7. The elasticity of aluminum foil is measured as increase of the

number of holes.

3.1 Theoretical work

The characteristic points of Swiss-cheese model are
1. The center of circles, spheres and so on, can occupy
arbitrary point in the space.
2. Overlapping area between circles in 2D space or over-
lapping volume of spheres in 3D can change continuously
from zero.
3. The number of the neighboring elements can change. Fig. 8. Measurement of electric current in the pure water, where a large
amount of rubber balls were inserted and the water flow through the
Most important property can be found in the critical ex- space of rubber balls was measured.
ponents of the transport coefficients, especially in the 3D
space. Estimated values of the critical exponent coefficients
such as the electric conductivity, the elasticity and the per-
meability are given in the Table 3 (Feng et al., 1987).
3.1.1 Conductivity and elasticity of 2D Swiss-cheese
model The critical exponents 1.1 for conductivity for a
foil of Al and 1.2 for a foil of Cu, and for elasticity 3.3
(Al) and 3.7 (Cu) were obtained in the experiment shown
in Fig. 7, where many holes are punched out in a sheet of
Al and Cu, respectively. The exponents for the conductivity
show good agreement with estimated values in Table 3, but
this exponent is the same as in the discrete model. The
exponents for the elasticity lie between the discrete one and
expected continuum one in Table 3 (Benguigui, 1984).
3.1.2 3D experiment It is difficult to build up appa-
ratus for 3D experiment of the Swiss-cheese model for con-
ductivity, elasticity and permeability. However, experimen-
tal apparatus for the Inverse Swiss-cheese model is easier
and was approximately built up by Miyazima et al., two of
them are shown in Figs. 8 and 9 (Miyazima et al., 1991,
1992; Okazaki et al., 1996).
We pushed many rubber balls into pure water in the
acrylic tube as shown in Fig. 8 and measured the electric
current from the top to the bottom through the water. If we Fig. 9. Measurement of elasticity of ensemble of rubber balls. The thin
press the rubber balls by a piston, the gap among the balls thread in the figure indicates an enamel wire which has much smaller
decreases together with current. If we measure the flow rate elasticity than rubber.
of water, we can obtain the permeability. The exponents
of the conductivity and the permeability were 2.4 and 4.4,
respectively, which show very good agreements with the ex- where the exponent of the discrete percolation model is 3.7
pected values in Table 3. (Maruyama et al., 1993).
Figure 9 shows the experiment for elasticity. First, we
put enamel wire into acrylic tube which has a very small 4. Concluding Remarks
elasticity. Between the enamel wire we inserted rubber balls The percolation model is the simplest one among mod-
softly. Initially the rubber balls had no contact each other. els appearing in physics. It is, however, a very important
When we push the system from the top slowly, then the model for a quantitative description of various phenomena
rubber balls began to contact and at some instant the rubber which appear in nature. The covering area of the percola-
ball system began to have elasticity between the top and tion model is not only materials but also social, economic,
the bottom. In this way we obtained the critical exponent biological problems and so on. Especially the universality
of elasticity 5.4 which agrees reasonably with expectation, is very important concept for estimating what the crucial
 Percolation S23

properties is in various phenomena. But, we know only few Jacobsen, J. L. (2014) J. Phys. A: Math. Theor., 47, 135001.
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