SCALING THEORY OF
PERCOLATION CLUSTERS

D. STAUFFER

Institut für Theoretische Physik, Universität zu KöIn, 5000 Köln 41, W. Germany

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 54, No. 1(1979)1-74. NORTH-HOLLAND PUBLISHING COMPANYSCALING THEORY OF PERCOLATION CLUSTERS
D. STAUFFER
Institut für Theoretische Physik, Universität zu Kdln, 5000 Köln 41, W. Germany
Received December 1978
Contents:
1. Introduction
I.!. What is percolation?
1.2. Critical exponents
2. Numerical methods
2.1. Series expansions
2.2. Monte Carlo simulation
2.3. Renormalization group
2.4. Inequalities
3. Cluster numbers
3.1. Scaling theory
3.2. Tests of scaling
4. Cluster structure
4.1. Cluster perimeter and internal structure

3
3
8
11
11
13
15
18
20
20
27
35
36

4.2. Radius and density profile
4.3. Droplets, ramification, and fractal dimensionality
5. Lattice animals
5.1. Scaling close to critical point
5.2. Exponents away from the cntical point
6. Other percolation problems
6.1. Random resistor networks
6.2. Modifications of percolation
7. Conclusions
Appendix 1. Evaluation of sums
Appendix 2. One-dimensional percolation
References

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D. Stauffer, Scaling theory of percolation clusters

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Abstracts:
For beginners: This review tries to explain percolation through the cluster properties; it can also be used as an introduction to critical phenomena

at other phase transitions for readers not familiar with scaling theory. In percolation each site of a periodic lattice is randomly occupied withprobability p or empty with probability I — p. An s-cluster is a group of s occupied sites connected by nearest-neighbor distances; the number of
empty nearest neighbors of cluster sites is the perimeter t. For p above p~also one infinite cluster percolates~through the lattice. How do theproperties of s-clusters depend on s, and how do they feel the influence of the phase transition at p =The answers to these questions are given by various methods (in particular computer simulations) and are interpreted by the so-called scaling
theory of phase transitions. The results presented here suggest a qualitative difference of cluster structures abo~’eand below pç: Above p~somecluster properties suggest the existence of a cluster surface varying as 5213 in three dimensions, but below Pc these “surface” contributions areproportional to s. We suggest therefore that very large clusters above Pc (but not at and below Pc) behave like large clusters of Swiss cheese:
Inspite of many internal holes the overall cluster shape is roughly spherical, similar to raindrops.
Forexperts: Scaling theory suggests for large clusters near the percolation threshold p~that the average clusternumbers n, vary as s~f(z),withz (p — p~)s”.Analogously the average cluster perimeter is t~= s (I — p)/p + s’ ifi
1(z), the average cluster radius R, varies as s~’ R1(z), andthe density profile D,(r), which depends also on the distance r from the cluster center, varies as s~ D1(rs°~,z). These assumptions relate theseven critical exponents a, ~, y, ~, v, a-, r in d dimensions through the well-known five scaling laws 2—a = y + 2~= $5 + $ = dv = j3 + I/cr =(r — l)/o~leaving only two exponents as independent variables to be fitted by “experiment” and not predicted by scaling theory. For the lattice“animals”.
2-f(z)i.e.
withthez ~number(a~— t/s)s”
g~,ofandgeometricallya~= (I — pj/p~.
different
All theseexpressions
cluster configurations,are variants
a modifiedof the generalscaling
scalingassumptiOnidea tor second-orderis derived: g,~s~t’/(sphase transitions+ t)’~~
that a function g(x, y) of two critical variables takes the homogeneous form xcG(x/yc~)near the critical point,with two free exponents b and c and as’~scaling function G of a single variable.
These assumptions, which may be regarded as generalizations of the Fisher droplet model, are tested “experimentally” by Monte Carlosimulation, series expansion, renormalization group technique, and exact inequalities. In particular, detailed Monte Carlo evidence of Hoshen eta!. and Leath and Reich is presented for the scaling of cluster numbers in two and three dimensions. If the cluster size s goes to infinity at fixed
concentration p, not necessarily close to p~,three additional exponents ~, 0, p are defined by: cluster numbers ~ sexp(—const . sC) and cluster
radii ~ s’. These exponents are different on both sides of the phase transition; for example ~(p <pa) = 1 and ~ >pc) = 1— lid was found frominequalities, series and Monte Carlo data. The behavior of 0 and of p(p <pa) remains to be explained by scaling theory.
This article does not cover experimental applications, correlation functions and “classical” (mean field, Bethe lattice, effective medium)
theories. For the reader to whom this abstract is too short and the whole article is too long we recommend sections 1 and 3.

1. Introduction

1.1. What is percolation?
What is percolation? Figure 1 gives a first answer: Percolation is a very fashionable field. The
number of papers published each year has gone up by a factor of 10 in ten years, due in part to theincreased interest in disordered systems generally. The greater part of the pronounced peak in fig. I
for the percolation publication rate could not be covered in the earlier reviews of Shante and
80.1
I

.

S

60

40

20

.
.

0

.
~i

.

S

•

S

S

I

1970
1975
Fig. 1. Variation with time of the number of articles published each year which carry the word peràolation, percolating etc. in the title. Tabulated
from the Permuterm Subject Index of the Science Citation Index and the Weekly Subject Index of Current Contents, Institute of Scientific
Information, Philadelphia, PA, USA. The critical exponent describing this curve has not yet been determined.
1965

4

D. Stauffer, Scaling theory of percolation clusters

Kirkpatrick [1],Essam [2],and Kirkpatrick [3].Among shorter reviews written since then we mention
a more mathematical presentation [4],a simple description of applications [51,and a lecture mainly on
random resistor networks [6].Kirkpatrick [7]recently gave a rather detailed review together with new
numerical data. About simultaneously with the present review other reviews are being prepared by
Essam [8]and by Pfeuty and Guyon [9],the latter emphasizing applications, which are largely ignoredin the present article.
These reviews usually emphasize the “bulk” behavior of the percolation system as a whole. The
present article reviews the scaling theory of percolation clusters, i.e. it concentrates on the often
neglected question of how these “bulk” properties can be explained by clusters, and how the cluster
properties depend on the cluster size. Taken together with ref s. [7—9]the reader will, we hope, get abalanced view of percolation. (Our list of references is an incomplete and subjective selection of
percolation papers.) Instead, if the reader does not have the time to read the present article fully werecommend that he restricts himself to our long introduction and to section 3 only. Then he isinformed on those cluster properties which at present seem no more controversial. De Gennes’s
review [5]is suggested for an elementary introduction into percolation theory and its applications.
What else is percolation? This review tries to show that, besides being fashionable, percolation is
also a very simple problem. It is easily defined (see below), and the methods of numerical solution are
also easy to understand in principle. In this sense the phase transition behavior of percolation, i.e. the“scaling” properties, can be used as an introduction to more complicated phase transitions and criticalphenomena [10,11]. In order to understand percolation it is not necessary to know what properties aspin has or what a free energy is. In short, one does not have to study physics in order to understandwhat is going on in this review. Basically, only the concept of probability and the purely geometricalcounting of “cluster” configurations is necessary; and usually we do this here on two-dimensionallattices like large chessboards. The most complicated mathematical methods used here (only seldomly) are Taylor expansions, the definition of the Gamma function 1’(x) = ~ y~~’ e~dy for realargument x, and the related knowledge of fff exp(—y2) dy = V~r.In fact, a simple percolation game,
which can be finished during one lecture, has been used repeatedly to give students an active
experience of what a critical exponent and a Monte Carlo experiment are [12].Of course, noteverything connected with percolation is simple: So far none of the critical exponents like 13 etc.,
which we will discuss later at the end of this introduction, has been calculated rigorously and exactly
for two or three dimensions, although for the two-dimensional Ising model of magnetism such exact
exponents are known since many years [10, 11]. Our review here will simply try to avoid thecomplicated problems and concentrate on the simpler concepts.
What is the definition of percolation? Imagine a very large quadratic lattice like a huge chessboard.
Assume that every single square on this lattice can be in either one of two states, which we call“occupied” and “empty”. Moreover, each square is occupied or empty entirely randomly, independent of whether its neighbors are occupied or empty. Thus the whole problem is defined, for agiven lattice type, by a single parameter p, where p is the probability (the same for each square) to beoccupied. Fig. 2 shows an example how a 20 by 20 chessboard is slowly filled up from p = 0.1 to
p = 0.9. If our lattice is not a chessboard (square lattice) we call its units the lattice sites; they are
either occupied or empty. In the triangular lattice, for example, these Units are thedots shown schematically in this diagram; in the simple cubiclattice they are cubes.
Numeroussuch latticeshave been studied [1,2]; the presently widespread belief in“universality”
asserts that it does not make much difference which lattice wechoose.

D. Stauffer, Scaling theory of percolation clusters

10

20

30

•..

.

:

/.0

~

5o~:.:.~:.

70!!

5

:!:~::!::.!i!:::‘..80::~::::::~:.::::.

601j~90

Fig. 2. Examples for percolation on a 20 x 20 square lattice, for p = 0.1, 0.2,. . .0.9. Occupied sites are shown as dots, empty sites are not shown.
The overlapping crosses at 60 percent probability give the largest “percolating” cluster.

In this percolation problem the occupied sites are either isolated from each other or they form
small groups of neighbors. These groups are called clusters:
A cluster is defined as a group of occupied lattice sites(1)connected by nearest-neighbor distances.
In fig. 2 the largest cluster at p = 0.6 was symbolized by overlapping crosses +, which distinguish it
clearly from the smaller clusters also present there. Isolated sites are regarded as clusters of sizeunity; and generally we call any cluster consisting of s occupied connected sites an s-cluster.
In a large lattice there will be more clusters than in a small lattice; thus it is convenient to divide thenumber of clusters by the number of lattice sites in the whole lattice. This ratio is called the number n,,of s-clusters if it is an average over many different distributions of occupied sites among the latticesites:
n5 is the average number (per lattice site) of s-clusters and depends on concentration p.

(2)

Eqs. (1) and (2) define the two most important concepts used in this report, the s-cluster and thenumber of s-clusters.
If p is close to zero, most occupied sites will be isolated, with only a few pairs and triplets present(fig. 2 with p = 10%). If, on the other hand, p is close to unity then nearly all occupied sites areconnected to each other and form one large cluster extending from one end of the lattice to the other(fig. 2 with p = 90%). According to present knowledge [1],in a sufficiently large lattice there is either
one or none, but never two or more such “infinite” clusters or “networks”. This infinite cluster
percolates through the lattice just as water is percolating through wet sand akng the network of wet
pores. Our examples at p = 70% and 80% also show that besides this percolating network many finite
clusters exist, too. A clear distinction thus exists for large lattices: Either an “infinite” cluster exists,
or it does not exist. Therefore it is plausible that for an infinite lattice a sharply defined percolationthreshold Pc exists, i.e. a critical point, where for the first time an infinite net*ork percolates throughthe lattice with finite probability. Thus Pc indicates a phase transition such that:

6

D. Stauffer, Scaling theory of percolation clusters

For p above Pc one percolating network exists;
for p below Pc no percolating network exists.

(3)

In this sense percolation is a phase transition [10,11] which generally can be defined as thephenomenon that a system exhibits a qualitative change at one sharply defined parameter value, if thatparameter is changed continuously. Only in an infinite system (“thermodynamic limit”) do we observea true phase transition in this sense. For example, in our finite system of fig. 2, we do not know
precisely whether p = 0.6 is above or below the percolation threshold p~for the large cluster
percolates horizontally but not vertically. Such accidental differences between horizontal and verticaldirections become less and less probable if the lattice size increases. Thus Pc is defined uniquely in an
infinite system. (Of course, even for p -~0we can produce an infinite cluster by filling merely one rowof our lattice with occupied sites. However, such a configuration has an extremely low probability fora large lattice in a truly random distribution of occupied sites. And we require above Pc to have an
infinite network in an infinite system with probability one.)More quantitatively, we call the percolation probability P the fraction of occupied sites belonging
to the infinite percolating network. Then P vanishes below Pc and is nonzero above Pc; close to Pc we
can define a “critical exponent” 13 by postulating P00 ~ (p PcY5 for p slightly above Pc. The behaviorof the infinite network and of large finite clusters, for p very close to Pc, is called the critical behaviorof percolation theory; the region of parameters where it applies is called the scaling region.
In general, every lattice site has three choices: It can be empty, with probability 1 p; it can bepart of the infinite network of occupied sites, with probability p P; or it can be part of one of themany finite clusters including single sites, with probability p (1 F
00). Since each s-cluster contains
exactly s sites, the probability of any lattice site to belong to an s-cluster is P~= s n, (remember thatn5 was defined as the number of s-clusters divided by the total number of lattice sites). The sum of allthese probabilities equals unity:
—

—

—

-

1 p + pP +—s n~= 1(4)

where ~ denotes the sum over all finite cluster sizes, s = 1, 2
Thus, if we know all the cluster
numbers n5, then we can calculate from eq. (4) the strength P of the infinite network. We see alreadyhere that the cluster numbers n, are the basic quantities for our discussion.
To study percolation experimentally with simple methods one can flip coins in a classroom
experiment [12]to produce easily a probability p = ~ for a site to be occupied. More efficiently, one
can ask a computer to do that, and for readers interested in Fortran programming a simple example ofsuch a “Monte Carlo” method is shown in table 1. A computer program similar to the one intable I produced the results for fig. 2; our example might be used for teaching Monte
Carlo methods since the execution time was smaller than the compilation time. (If computer
experts ask for money to make such Monte Carlo computer simulations they in general
will not (at least not officially) use that money on the roulette tables of that Mediterranean
town in order to produce a series of random numbers needed for percolation. Instead, the computerhas built in methods to generate pseudorandom numbers by multiplying large numbers and dropping
the leading digits. For the CDC computer used here, this aim is achieved by the function RANF(n)which gives, for arbitrary n, a real number distributed randomly between zero and unity. The program
compares this random number with the concentration p to decide whether the lattice site is occupiedor empty.)

D. Stauffer, Scaling theory ofpercolation clusters

7

Table I
A simple Fortran computer program to produce a percolation example on a 20 x 20 square lattice. The function RANF for this computer
produces random numbers distributed evenly between 0 and I.
Empty places are stored as zeros, occupied sites as ones.

2

DIMENSION IS(2Ø,2~)
P =N=1
DO 1 1 = 1,2~
DO 1 J = 1,20
IS(I,J) = 0
IF(RANF(N).LT.P) IS(I,J) = 1
DO 2 I = 1,20
WRITE(6,3) (IS(I,~J),J=1,20)

3

FORMAT (1X,2012)

1

STOP
END

Besides such computer experiments, also real experiments can be done to study percolation.
Imagine for example that each occupied square on the chessboard is electrically conducting, whereas
empty sites are insulators. A current can flow from one conducting square to the other if both squares
share one line as common border. This means that an electric current can flow only within one cluster,and not from one cluster to the other. Only if a percolating network is present, i.e. for p above thepercolation threshold Pc, has the lattice a nonzero bulk conductivity. In this form the percolation
problem is known as the random resistor network; we may also interpret it as oil or water flowing
through porous rock or wet sand. By coating candy dragees or plastic bul1~tsfrom toy guns withcopper, the Marseille group [13]studied the conductivity above Pc. Random mLxtures of these sphereswith equal size but two different colours arrange themselves into a triangular~lattice if shaken softly
on a flat surface; then clusters and the infinite network can be seen easily. Unfortunately, as
quantitative experiments these methods are too complicated [13]for teaching purposes. Earlier simple
experiments included the punching of random holes into conducting paper by Last and Thouless [14],and the random cutting of wires in a steel-wire mesh, bought in a local hardware store, by Watson andLeath [15]. But in the present review we will be satisfied with regarding percolation as a mathematically well-defined and simple problem; the difficulties of approximating rea1isti~disordered systems innature by percolation models and the problems of conducting experiments with real materials are left
to other reviews [5,9].
Readers who insist that everything is explained to them in the language of the spin ~Ising model for
ferromagnets at finite temperature T in a magnetic field H can also be satisfied now; other readers
may go ahead to section 1.2. Imagine that in a lattice, magnetic atoms and ponmagnetic atoms are
mixed randomly and then cooled down (“quenched”) to very low temperatures. Then the magnetic
atoms form clusters according to percolation theory, with p = mole fraction of the magnetic spins.
Each spin has a magnetic moment ~ pointing either up or down. Nearest-neIghbor exchange forces
force the magnetic moments of neighboring atoms to be parallel since the thermal energy at these lowtemperatures is too low to break up these exchange bonds. Thus all moments within one cluster are
parallel. (We neglect any dipole—dipole interaction between the spins.) The whole cluster therefore
acts as if it would be one large isolated (“superparamagnetic”) spin. An isoiated single spin has inthermal equilibrium a magnetization of tanh(~H/k8T)in units of the saturation magnetization. Thus
for a superparamagnetic cluster the argument of the tanh is replaced simply by s ~HIk8T since its

8

D. Stauffer, Scaling theory of percolation clusters

magnetic moment is s~s.The contribution of clusters to the total magnetization moreover contains afactor s since a large cluster contributes more than a small cluster. Finally, the contribution of all
s-clusters is proportional to the cluster numbers n~.Therefore, the total magnetization contribution
from finite clusters is (1/p),i~tanh(s~H/kBT)in units of the saturation magnetization. The
infinite network contains the fraction F00 of all spins and is oriented either fully up or fully down. Thus
the whole equation of state M = M(p, T, H) for the magnetization M (units of the saturation
magnetization) is given by
M = ±P00+(l/p)’~n~tanh(s~H/kBT).

.

or, with the use of eq. (4):pM= ± ~.

n~)+~
ç ~ tanh(s~LH/kBT).
.

(5)

In this way for low enough temperatures the magnetization in this “dilute” ferro-magnet is related
directly to the cluster numbers. For p below Pc no infinite network is present, and for H = 0 the finite
clusters give zero magnetization, corresponding to the paramagnetic state. But for p above Pc theinfinite network gives a “spontaneous” magnetization P~,i.e. a “remanence” even for zero magnetic
field. Eq. (5) looks as if it is a direct experimental realization of the percolation problem; but actuallyexperiments as a function ofp are quite difficult [16],and not much information has been gained so far
from such experiments on the cluster numbers. The time-dependent behavior of this model [61]is notdiscussed here. [Expertson metastable systems may notice that the ± sign in eq. (5) gives above p~,aunique and smooth but nonanalytic continuation of the stable branch, M parallel to H, into theunstable branch, M antiparallel to H. There are no van der Waals loops and no spinodal lines with
8MIDH = ~ in this expression for p above p~.]
These remarks conclude our qualitative description of percolation; we now turn to the quantitativedescription of the scaling region by critical exponents.
1.2. Critical exponents
The behavior of systems close to a phase transition [10,11], also for percolation, is usuallydescribed by critical exponents a, /3, y,... (except for first order phase transitions). For our review weneed only the definitions[~

2~n~(p)Jsingsor p —p~I[~s.n(p)][~[~oc(p—p~)~or p —p~fl~2,~(p)]

sing

ç . n

s

~(p)

(6b)

sing

5(p)e~5]or h’~
singor Jp —p.fl.

(6e)

All five equations are supposed to be valid only for p near p.~or h near zero. For simplicity we assume
that the same exponent describes the behavior above and below Pc, when appropriate. The subscript

D. Stauffer, Scaling theory ofpercolation clusters

9

sing denotes the leading nonanalytic part of the subscripted quantity and does not necessarily meanthat that quantity goes to infinity; for example, if close to p~a quantity A varies as A =then the “singular” (or critical) partA0+A1(p —p~)+A2(p—p~)’355+A3(p—pCf855+A4(p —p~)2~
Asing is the term A
355,whereas A
2 are denoted as the “analytic
0+A1(pto—pC)+A4(p—pj review will concentrate on the
background” and A2(p —p~Y 855
as a “correction
scaling”. Thisp~~ to this definition we have ~ S fls]sing or P.. in eq. (6b), since thesingular parts only. 3(p(According
concentration p which also appears in eq. (4) contributes only to the analytic background etc.) The
real number h in eq. (6d) is merely a dummy variable to calculate a sum and approaches zero; thebackground of that sum is its value at h = 0 which is ~ s n,(p~)= PC~from eq. (4). Finally, ~ is thecoherence length (or correlation length, connectivity length) and corresponds to the average radius ofa typical percolation cluster, as will be discussed later. We neglect here the possibility that logarithmic
factors appear besides powers of p Pc; in six dimensions such factors are important where e.g.
[~.s2n,]~2u17 according to Essam et al. [17].(Another exponent ~ 2— yiv isohere
r (PC [18].)
pY’Ilog(p~ p)1
not needed1~
So far eqs. (6) are merely definitions of exponents by various moments of the cluster size
distribution n
5. But these definitions are indeed analogous to what is usually defined at other phasetransitions where T~ T replaces p Pc as the distance from the critical point. In eq. (20) we will see
later that ~2~
corresponds
to a infree
energy,at thermaltransitions
assumedto varytheasin,n its
singular part,
order
that which
the specificheatphasedivergesas (T isTCI”.
Moreover,(TT~)
percolation probability P
0, corresponds to the spontaneous magnetization or (TC T)~,similar to ourremarks after eq. (5) for dilute 2n~,ferromagnets.
We phasealso seefrom eq.
that the susceptibility
and at thermaltransitions
this(5)susceptibility
diverges asx(T 3M/aHT~)~
as
is connecteda generalization
with ~ ofs the Curie—Weiss law. At the critical temperature T T~themagnetization is usually assumed to vary as H”5, analogous to eq. (6d); indeed the factor e~in eq.
(6d), basically introduced by Gaunt and Sykes [19],can be interpreted as a “ghost” field (with e~
replaced by 1 h), a magical trick avoided in this review but used elsewhere [20—23].
Anotherinterpretation of e is given in ref. [24]in connection with semiconductors. Also a coherence length
can be defined at thermal phase transitions to describe the decay of correlations over long distances,diverging as IT TC~0,similar to eq. (6e). Since about ten years this exponent notation is standard forphase transitions and has been adopted consistently in the literature also for percolation. Of course, areader not familiar with thermal phase transitions may simply overlook the above analogies.
A complete understanding of percolation would require to calculate these exponents exactly andrigorously. This aim has not yet been accomplished, even in general not for Other phase transitions.
(Ref. [25]gives reasons to hope for an exact result in two dimensions: v = log(3”2)/log(~).)The aim ofa scaling theory as reviewed here, is more modest than complete understanding: We want merely toderive relations between critical exponents. And since this review deals mainly with the scaling theoryof clusters, we want to understand how the above exponents like /3 can be calculated from cluster
properties and cluster exponents.
For even in the cluster properties some simple power laws are found. Fig. 3 gives Monte Carlo
results for the cluster numbers n
5 at p = Pc = ~ in the triangular lattice. The data in~this double
logarithmic plot follow a straight line with slope —r —2, i.e. log n5 = const r log s, or n~or s~.
What relations exist between this new exponent i~and the other critical expoilients defined in eq. (6).
Does a single exponent like r suffice to describe the cluster numbers in the scaling region? These are
the questions our review tries to answer. Of course, to test these scaling theotlies we will have to use
the numerical methods and results for finding critical exponents, as described in greater detail in
—

—

—

—

—

—

—

—

—

—

—

—

—

10

D. Stauffer, Scaling theory of percolation clusters

1

I

I

1015

0

I

102

I

I

10~

106

Fig. 3. Computer simulation [26] of cluster numbers in a 4000 X 4000 triangular lattice at the percolation threshold. The straight line showsn,(p0)xs~,i.=2section 2 and as evident already in this example of fig. 3. After we have explained in section 3.1 thescaling theory of percolation clusters we will list there the present estimates for the exponents andalso for the percolation thresholds Pc. We merely mention here that for two dimensions, we haveroughly
a= —0.7, j30.14,

y2.4,

818,

i’1.35,

results which are expected according to the universality hypothesis to be valid for (nearly) all
two-dimensional percolation problems, independent e.g. of the lattice type. On the other hand, p.~depends on the lattice and equals 0.50 for the triangular and 0.59 for the square lattice.
[Oneexactly solved problem is percolation on the Bethe lattice, also known as Cayley tree [27—30],
where a = 1, /3 = 1, y = 1, 5 = 2. We refer to refs. [2,30] for a discussion of Bethe lattices andrestrict ourselves here to the not yet exactly solved cases where the exponents are not integers.]
In this sense the percolation problem can serve as an introduction to the scaling theory of phasetransitions and critical phenomena: the critical exponents are defined by mathematically simple
expressions in eq. (6); and later in section 3.1 we will derive relations between them [31],using asimple scaling ansatz for the cluster numbers n,. As promised before one does not have to know
concepts like “free energy” of “magnetic moment” to understand the definitions (6), contrary to mostother phase transitions [10].
But besides being perhaps the simplest not exactly solved phase transition, percolation theory also
serves as an introduction to cluster approximations of collective phenomena. In statistical physics,
systems with interactions between the units (molecules, spins,...) in general cannot be calculated
exactly, whereas most systems without interaction are easy to solve. Thus an often-used approximation is the cluster (or droplet) approximation which tries to transform the problem ofinteracting units into the approximation of non-interacting clusters. For example, to describe theequation of state and the condensation process (nucleation) in a real gas, one may group the gas
molecules into clusters of neighboring molecules (“liquid droplets”) such that interactions between
different groups can be neglected. Interactions within the same group are taken into account either
exactly (for very small clusters) or by simple approximations containing only a few free parameterslike surface tension etc. If the critical point of a fluid is described [30]by such a “droplet” model, it is—D. Stauffer, Scaling theory of percolation clusters

11

characterized by the appearance of very large clusters leading to critical opalascence in lightscattering. Such cluster models exist not only for fluids but also for magnets, polymers, structuraltransitions, semiconductors. For early references see Domb [32].
To put such cluster models for general phase transitions onto a more rigorous basis one can study
simplified models, instead of real water droplets, to look directly at the micrOscopic clusters and theirimpact on macroscopic quantities like susceptibility etc. In order to be uselul such a model shouldhave five properties:(1) it can be used in three dimensions(ii) for every configuration there is an unambigous and unartificial separation of units into clusters(iii) the definition of clusters according to (ii) is simple enough for computer handling(iv) at least some macroscopic quantities can be calculated reliably from the cluster properties(v) the system has a critical point; there and only there large clusters appear.
Some years ago there was reason to hope that the Ising magnet (lattice gas model) fulfilled all theabove requirements, since in two dimensions computer simulations [33]agr~edwell with the Fisher
droplet model [31,34], if a “cluster” is defined as a group of overturned spins connected bynearest-neighbor exchange forces. Unfortunately later work [35] showed that condition (v) is violated
in three dimensions. Thus the cluster situation in this Ising model is rather complicated [36],and
percolation seems right now to be the only problem where all the above requirements are fulfilled.
Thus percolation can be used as a guide to clustering phenomena at other phase transitions. Of course
there is no guarantee that other phase transitions will have the same cluster properties as the simple
percolation problem; but it seems plausible that a complete understanding of simple clusters (i.e.
percolation) is helpful for a better understanding of more complicated clusters (e.g. fluids, magnets). It
should be noted, however, that the above difficulties with clusters at other phase transitions arise
mainly near the critical point; far below the critical temperature of a fluid, for example, an often
experienced example of quite well-defined clusters are raindrops; they are nothing but grown-up
stages of those clusters relevant for the liquid-gas phase transition in a supersaturated vapor [37,38].
The possible similarity of percolation clusters with raindrops will therefore be discussed in detail, seesection 4.3.1.
2. Numerical methods
This section describes shortly the main methods used so far to calculate percolation quantities
approximately. It is not necessary to read this section in order to understand (we hope) the laterscaling theories, where the results of these numerical methods are reported and interpreted.
2.1. Series expansions

The average number n, of small clusters can be calculated exactly. Take, for example, a pair in thesquare lattice, i.e. an s-cluster with s = 2. It consists of two occupied sql$ares surrounded by six
empty neighbor squares. Moreover it can be oriented either horizontally or~vertically in the lattice.
Thus the average number of pairs is n2 = 2p2(1 p)6 since in percolation eacl~of the squares involved
is either occupied (with probability p) or empty (with probability 1 p) entirely independently of theother squares. Generally we denote as the perimeter t the number of empt~’lattice sites which arenearest neighbors of occupied cluster sites, and as g~,the number of geom~tricallydifferent cluster
—

—

12

D. Stauffer, Scaling theory of percolation clusters

configurations of s sites with perimeter t. (The g,~are also called the number of “animals”; see section
5; if two configurations are identical apart from a rotation they are counted as different; if they are
identical apart from a translation they are counted as the same configuration.) Then we have exactlyn,(p) = ~ g,,p’(l —p)’.

(7)

The following diagrams show small clusters on a square lattice together with the resulting expressions
for n,; the cluster sites are denoted by dots, the perimeter sites by crosses:
x

x

Xx

X•X

X.X
X

X..X
XX

X.X

XXX

X-X
X.X

X...X

x

XXX

x

+

x

~x

xx

XX
X~.X

XX
X..X

X~~X
X.X

X~X
X~X

XX

XX

X

X

X.X
x

n, = p(l pY’; n2 = 2p2(l p)6 n3 = 2p3(l p)8 + 4p3(l p)7.
This method of exact cluster counting was already used twenty years ago [39—41].
The above
diagrams show that for larger cluster sizes more possibilities and thus more difficulties arise. The
exact result for s = 14 in the triangular lattice, with q = 1 p, is [42]
—

—

—

—

—

n,

4(3q’6+ 168q17+ 1524q’8+ 10029q19+46119q20+ 185220q2’ +605766q22+ 1730943q234(p) =p’+4287699q24+9131949q25+ 16871550q26+26571525q27+35061399q28+3796541q29+ 32198928q3°+ 19012074q3’ + 5812482q32),

a result which illustrates convincingly that the counting is done best on a computer [43].
Even then then~are known only up to 5 = Smax, with Smax usually of order ten. For example we have Smax = 17 in thesquare lattice, 5max = 11 in the simple cubic lattice, and SmaX = 7 for the hypercubic lattice in arbitrarily
large dimensionality [44,45]. For applications, a table of n, (p) in two and three dimensions calculated
from these polynomials was given in ref. [46].
For the triangular lattice, besides the full range of g,.~at fixed s up to 5max = 14, exact g,~for fixed twere given in ref. [47]up to tmax = 22. Duarte [48]analyzed in detail the behavior of averages atconstant perimeter t, in particular the deviations from a Gaussian distribution. We will not use thisinformation here since we will look at the distribution of perimeters t at fixed size s instead.
These exact results of Sykes et a!. [42,47]have the advantage that they can be used even far awayfrom Pc where the cluster numbers n, become very small. That region, for example, is important for
nucleation applications [38,49]. Moreover they give not only the cluster numbers but also the average
perimeter t, of s-clusters: t, = ~., t g~,p’(1 p )t/~, gS (1 p )t~ Their main disadvantage is theirrestriction to small and intermediate cluster sizes s. So far most extrapolations to s
[46,50, 51]
employed rather simple methods: If a quantity A, is expected to have a finite limit A
00 for s —~co, thenA, is plotted against 1/s. If one is lucky these data follow a straight line for large s; the intercept(1/s = 0) of this line gives A00. If instead a clear curvature is seen, one plots A, against some otherpower of 1/s until the points fit a straight line; again the intercept of that line gives the extrapolated
A00. Particularly accurate results were given in ref. [52].Usually the resulting extrapolations are about
as accurate as those from Monte Carlo results, as can be seen by comparing the corresponding resultsof refs. [26,51] or [53,52].
Better methods of extrapolation exist to calculate the 2n,
criticalexponents
for thevariousor (p~p)~belowp.~,eq. (6c).moments
Instead ofoftheclustersize
distribution,e.g.
theexponent
‘y
for
~,
s
first extrapolating for large s the cluster numbers n~,one deals directly with this sum and expands it
into a power series in p by using the exact expressions for n,(p) and by collecting the appropriate
.

—

—

—~

—

D. Stauffer, Scaling theory ofpercolation clusters

13

Since n, varies asp’ apart frompowers of p arising from pS(1_p)t in eq.(7): ~52fl,(p)=>~kakph1.
higher powers in p, exact knowledge of n, up to s = 5max also gives the exact expansion coefficients ak
up to k = 5max additional tricks [42]gave ak for k = Smax + 1 and Smax + 2. These series expansions in
powers of p etc. have given this subsection its name; for simplicity we will refer to all results derived
from the exact n,(p) as “series” results even if no power series in p was used. The analysis of such
power series in terms of critical exponents is a standard method of phase transition theory and isreviewed in vol. 3 of ref. [11];usually exponents estimated in this way are ipore reliable than thoseestimated by other methods. A rather simple way of series analysis is the “ratIo” method: if ~k akp is
supposed to diverge as (Pc p)~’and if the ratios ak+ 1/ak are plotted against 1I(k + 1), then the data
should follow a straight line for large k, with intercept i/Pc and slope (y 1)/Pc, as the reader cancheck by himself, employing the Taylor series f(p) = ~k [dkf/dpt(]~.opkIk!More complicated but
usually more accurate are the Padé approximations (particulary when the ratiO plots oscillate), where
the power series is approximated by the ratio of two polynomials in p [19,44, 54, 55]. The accuracy
sometimes can be increased further by suitable transformations of variables [~6].Practical experience
with such series methods has accumulated over many years of experience with other phase transitions[11].
An important difference between percolation and other phase transitions is,~however, that no exact
expressions so far have been published for the cluster numbers n,(T) in, say, the Ising model ofmagnets. Only power series of thermo dynamic quantities like susceptibility 4tc. as a function of lITor exp(—2p~H/kBT)have been derived and analyzed, e.g. in ref. [57]. As mentioned above, percolationis the simplest phase transition as far as clusters are concerned.
——

2.2. Monte Carlo simulation

Examples of Monte Carlo computer simulations were already given in the~Introduction, figs. 2, 3
and table 1. Similar to series expansions, the quality of the data is limited by the number of computer
hours available (and also by limitations in the computer memory). But whereas “series” give exact
expressions if s is near 10, Monte Carlo results are available for much larger clusters, s i03, withfinite error bars. (See ref. [58]for a theory of Monte Carlo fluctuations ii~the cluster numbers.)
Therefore it is seldomly necessary to extrapolate Monte Carlo results to s -+oo, since s = 102. iO~isusually close enough to infinity, if the finite error bars of the computer experiment are taken into
account. If possible, a combination in one plot of exact series results for small s and approximate
Monte Carlo data for large s gives a good impression of the accuracy and limitations of both
approaches. (We refer to Binder’s book [591for reviews of thermal Monte Cailo methods.)
An advantage of percolation compared with other phase transition is that ~on-equiiibriumdilficulties do not appear in most percolation studies: If a whole lattice is filled once With sites, the resulting
clusters can already be used for the statistics of cluster numbers etc. But for thprmal phase transitions
one first has to simulate perhaps thousands of Monte Carlo steps per site until the system has comeclose enough to thermal equilibrium in order that the averaging procedure can start.
Our presentation here is restricted to three different methods: SimuIation~of the whole lattice;
growth of one cluster; and shape fluctuations for one cluster.
—~. .

2.2.1. Simulations of the whole lattice
Table 1 gives the usual method of filling a lattice of N = L” sites in d dim~nsionsrandomly withpN occupied sites. In a thorough study one should vary the length L of the lattice and extrapolate the14

D. Stauffer, Scaling theory of percolation clusters

results to L = ~ [60,61].So far the largest systems studied seem to be L = 4000 in two and L = 400 inthree dimensions [26,61];care with the computer memory is essential for such large systems. Each ofthese lattices can be simulated once or several times; ref. [62] simulated a 110 X 110 lattice forty
thousand times. “Finite size” effects due to the limited size L can be annoying and might be reducedby employing periodic boundary conditions where layer number L + I in a cube is identified with thefirst layer to give a quasi-infinite lattice. The really complicated programming problem is the countingof clusters; orders of magnitude in computer time can be saved by efficient algorithms. We refer thereader to refs. [7,63, 64] for these details.
Monte Carlo simulation of percolation lattices were done already twenty years ago [65];for thecluster numbers n, the extensive tables of Dean and Bird [66]played, after a long time lag, an
important role in the development and testing of scaling theories [67—70]
and became obsolete only
quite recently [62,71, 72, 26]. The method has been used to calculate cluster numbers, perimeters,
radii and energies. (The energy is the number of occupied—empty bonds for the cluster.)

2.2.2. Growth of one cluster

Leath’s method [73]divides the lattice into concentric shells with the thickness of a nearestneighbor distance. Starting with the innermost site occupied, all lattice sites in the next shell are
randomly either occupied with probability p or left empty with probability i—p. It is tested which ofthese newly occupied sites are connected with the cluster in the inner shells. Then the next shell isfilled randomly, and the process is repeated. It stops when in one new shell no sites were occupied
which are connected with the cluster formed earlier in the inner shells. This method produces clustersof size s with a probability s n,, i.e. with the probability that the origin belongs to an s-cluster. In themethod of section 2.2.1 the formation probability is proportional to n, instead. Thus Leath’s methodenhances the statistics for the large clusters needed for a scaling analysis. Also there are no “finitesize” effects in Leath’s method since L is infinite. But the method requires a cutoff Smax in the cluster
size, taken as about i03, and thus could not give results like fig. 1 where 5m~ was near 106. And since
this method is inefficient above p~the results so far [71—73]
were restricted to p ~ PC. Detailed tables ofcluster numbers were made available in ref. [72].The method has been used to calculate cluster
numbers, perimeters and radii.

2.2.3. Shape fluctuations for one cluster
If one is interested not in the number n~of clusters but in other properties like-cluster radil etc.,
one can start with a typical shape of a large cluster and try randomly any changes in the cluster shapewhich do not change the cluster size s and do not separate the cluster into several parts. If theseattempted random changes are adopted with a probability (1 ~where t~tis the change in theperimeter t produced by a given change of the cluster shape, then one approximates in this way thetrue distribution of cluster shapes according to eq. (7) at fixed s. (Here one has to wait fornonequilibrium effects due to the initial configuration to die out.) In this way, analogous to an early
Ising model study [74],Stauffer [75]looked at the average perimeter and radii for s near 102; density
profiles were analyzed in ref. [76].No information on n, can be gained in this way, but the method canbe applied to large clusters far away from Pc where the other Monte Carlo methods fail. In particular,
the “animal” limit p =0 can also be treated with this algorithm.
—

D. Stauffer, Scaling theory ofpercolation clustersIS

2.3. Renormalization group
Only since 1975 has percolation been investigated [68] with Wilson’s renormalization group
technique which proved so useful for thermal phase transitions (vol. 6 of ref. [11]).Basically twovariants of that method have been used for percolation: The epsilon expansion and the real space
renormalization. Only the second method is explained here fully, and only a few papers are cited.
Kirkpatrick [7]reviews renormalization technique in greater detail.
The basic idea behind renormalization is the assumption, eq. (6e), that near p~the coherence length~ diverges. If we look at any clusters of diameter 1, with I much larger than the nearest-neighbordistance of the lattice, then we have in general to distinguish between I <~ and I > But right at Pc
we have ~ = ~, and all finite lengths like cluster diameter are much smaller than Thus it should not
matter at Pc on what a length scale we are investigating the system; apart from simple scale factors thesystem looks similar whether we look at it with the eye, with a magnifying glass, or with an optical
microscope (assuming the nearest-neighbor distance to be lAngstrom, visible in an electron microscope only). Thus we may “renormalize” the system by not looking at each lattice site separately but
by averaging over regions
which have a diameter of b lattice constants. Then we may averageagain over these averaged regions, with the same length factor b, etc. At p = Pc the averaged regions
are still much smaller than the coherence length even after numerous such averaging iterations; andnothing drastic should have changed due to this renormalization. In this sense the critical point Pc is amathematical fixed point of the renormalization transformation. If this transformation is done on thelattice itself it is called real space renormalization; if it is done in Fourier space it leads to the epsilon
expansion. Scaling is built in both approaches; and the main sucesses have been with the calculationof critical exponents like ii.
~.

~.

‘~

2.3.1. Real space renormalization
The above ideas have been made quantitative in many papers [7]; our presentation here
follows ref. [77] for the triangular lattice. (We thank P.J. Reynolds for his patience in explaining
this section to us.) Imagine that all lattice sites are put together into triangles
in such a waythat each site belongs to only one triangle. The center of each triangle is replaced by a new“supersite” 0 which represents the average (in the renormalization sense) over the three original sites.
The whole lattice then looks like fig. 4, where the small triangular A symbols show the “super-supersites”, i.e. the averages over three supersites 0. We see that the supersites 0 and the super-supersites
A in that figure form each a new triangular lattice, similar to but larger than the original triangularlattice formed by the dots We can continue by representing each separate triangle of A by a new
symbol *, these stars would again form triangles, etc. For simplicity these higher orders of therenormalization transformation are not shown in fig 4. In every case the lattice constant of the newlattice equals the lattice constant of the old lattice multiplied with a factor b = 3hI2~
If p is close to but not identical to PC we have a large but finite correlation length ~ which,expressed in centimeters, is the same in the original lattice and the renormalized superlattices. In theoriginal lattice we have ~ = ~p pcI” where ~ is of the order of the nearest-neighbor distance. Thusin the superlattice after one renormalization step we have ~ = ~p’ P~V,~ = b ~, where ~ is of
the order of the enlarged nearest-neighbor distance in the superlattice. The difference between therenormalized p’ and the original p takes into account that only at p = p’ = Pc the two lattices arecompletely similar to each other. Equating the two expressions for ~ we get b IP’ PcI~= IP PcI0,, orv = log(b)/log[(p’ p~)/(p Pc)].
(8)
..

—

—

.

—

—

—

—

-:

-:

-:

16

D. Stauffer, Scaling theory of percolation clusters

0

0

•0~
A

0
•

0

0

0
A

oA

A

0

0•
0

•0•
0

A

•0•
A

0

0
0

0

0

•

0

0

0

o
0

A

A

0
A

o
0

0

o
0

0

0

•0
A

o

0

0

0

o

•0~
A

0

o

A

0

0

o

A

0

0
0

A
0

0

0
A

0

0

0

•0

0

Fig. 4. Real space renormalization on a triangular lattice. At each iteration, three sites are combined into one supersite, denoted here by adifferent symbol. This picture was produced by S. Kirkpatrick on his computerized typewriter [7].

How do we define if the renormalized supersite in the triangle a is occupied? If all three sites ofthe original triangle are occupied (empty) then also the supersite representing the average of the threesites is occupied (empty). If only two sites are occupied, we define the supers ite as occupied, but if
only one original site is occupied, the supersite is defined as empty. Our diagrams show schematicallythe correspondence between occupied
and empty (x) sites in the original lattice and thesuperlattice:
.

()

—.

—

.
‘

X

=..
.

X

‘

X
X

•
‘

X

X.

XX

This correspondence gives the relation between the concentration p of occupied sites in the originallattice, and the probability p’ that a supersite is occupied. For example, two occupied sites can
produce an occupied supersite in three ways:X; each possibility occurs withprobability p2(1 p). Thus in total the probability p’for a supersite to be occupied isp’=p3+3p2(l—p).
.

.

•

—

The fixed point p” is determined by the nontrivial (p0, p1) condition p = p’ = p* and agrees withthe critical point Pc
Pc~

since then p’ = ~+ 3 ~ =b = 31/2 gives= p. Expanding for small p ~~Pc we get p’—p,~= ~(p ~Pc).

v = log(3112)/log(~)= 1.354 755 65.

Thus eq. (8) with
(9)

This example showed rather easily how critical exponents are calculated by renormalization technique. Actually in this example not only is p, = ~the desired exact result but also the exponent v in eq.
(9) is hoped to be exact [25]. This result of Klein Ct al. [25] (see also Reynolds et al. [22]) would be theD. Stauffer, Scaling theory of percolation clusters

17

first exactly calculated percolation exponent, if their argument could be made rigorous. At least it
agrees within a few percent or better with results derived from other methods, as cited in ref. [25].
Similar methods can be applied, although less successfully, to other lattices. One may also averageover cells larger than the triangles with only 3 sites discussed above. Very accurate results were
derived by calculations for very large cells (up to b = 500) of the renormalization transformation withthe help of Monte Carlo methods by Reynolds et al. [78].They checked if the~cell contained one large
cluster extending from one side to the opposite site, and then called the supersite for that cell
occupied. Similar to the pure Monte Carlo results of the preceding subsection, these renormalization
results were then extrapolated to b
giving e.g. v = 1.356±0.015,in e*cellent agreement witheq. (9) and with an accuracy comparable with or better than series expansions [18].Further work withthis combination of Monte Carlo and renormalization methods is in progress.
These methods gave properties of the percolation lattice as a whole but not directly on clusters as afunction of cluster size. A first attempt in that direction was made by Kunz and Payandeh [79]and,more quantitatively, by Kinzel [80].Kinzel [80]calculated the cluster numbe~sfor all p between zeroand Pc and thus showed that real space renormalization works also far away from the phase transition.
—*~,

2.3.2. Epsilon expansion
Another renormalization group method consists in expressing the fluctuations in a system by
Fourier components. Then one sums them up over all wavevectors iteratively, just as we averaged
iteratively over the real-space triangles above. An elementary review of that method was given by
Wilson [81].The summation over the wavevectors cannot be done exactly, and perturbation methods
in terms of a small parameter are used, similar to diagram techniques in the field-theoretical
approach to quantum many-body problems.
For percolation, the epsilon expansion [68]first transformed percolation into a thermal phasetransition in the socalled Q-state Potts model [21] for Q 1. Kasteleyn and Fortuin [211,as reviewed
already by Essam [2],derived for the partition function Zp05~.of this Potts model:—*

L~Potts—/2m,

where each lattice site can be in one of Q different states. Here ~m, = L m, is the sum over the actual
cluster numbers m, in a given configuration, and n, = (m,) is the average, cluster number. Theseaverages (...) are defined over all random (bond) percolation configuration~,where temperature T
and concentration p are related by log(1 p) or 1/T. For Q = 2 the result cc~rrespondsto the spin ~Ising model of ferromagnets. We see easily that the limit Q 1 means percolation. For then we
expand Q~” = exp[ln(Q)~,m,] 1 +ln(Q)~,m, 1 + (Q 1)~,m,; thus —Fp0555/kBT In(Z~0515)=
In(1 + (Q l)~,m,) (Q 1)).~,(m,), where F~011,is called the free energy:
—

—*

—

—

—

F~011,——k8T(Q—1)~n,for Q—~l.
In this sense the sum over all cluster numbers in eq. (6a) corresponds to the limit of a Potts model freeenergy (suitably normalized). Thus a solution of the thermal phase transition~in the Potts model also
gives a solution for the probabilistic phase transition of percolation. (The linüt Q 0 corresponds totrees percolating on a lattice [21,82].)
With these methods the perturbation expansion to order and 2 with
6— d >0 gives in ddimensions [83]:—~18

D. Stauffer, Scaling theory of percolation clusters

p = 1 —17— (61/12348)e2
y = 1 + 17 + (565/12348)2

+..

(lOa)
(lOb)

•

+.~

with the other exponents following from the scaling laws given later in section 3.1.
The crucial step was hidden before eq. (lOa): Our perturbation parameter basically is the dimensionality d. What sense does it make to use in this way non-integral dimensionalities like d = 5.9 where
eq. (10) should be a good approximation? (Later we will mention why six dimensions play a specialrole for percolation [84].)Obviously what we really need are results in three, and perhaps two,
dimensions. But numerically already the rather short series in eq. (10) give reasonable results if
applied to three dimensions ( = 3, p = 0.53, y = 1.84), exponents which are wrong by 0.1 to 0.2 only.
More importantly, later we will make scaling assumptions which are asserted to be valid for all
dimensions (below 6). If one can prove such an assumption to be wrong in the epsilon expansion ofeq. (10), then there is little reason to believe it is valid generally or in three dimensions. The
introduction of non-integral dimensions by Wilson and Fisher [85] is a new mathematical tool todescribe reality, just as the introduction of irrational and imaginary numbers was such a new concept
many centuries or millenia ago. We simply have become accustomed to these “imaginations”.
Certainly, if L” is the volume of hypercubes in d dimensions for all integer d, then L” is a plausible
definition for the volume in general nonintegral dimensionalities d. Only applications of that concept
to real problems can show if it is useful; and now we give two examples of such applications:(i) In polymer physics it was long assumed that excluded volume effects for self-avoiding walks
were described by the Flory formula: average polymer radius or (length)0, for large polymers, withv = 3/(d +2) = ~+ ~(4 d) +.... But de Gennes [86] showed that the epsilon expansion gives v =+ ~(4 d) +...
proving that the Flory formula cannot be exact in general.
(ii) For percolation, in some expressions correction factors of the form I + O(s2~T_1) with o =1/(13 + ‘y) only slightly smaller than ~were derived, eq. (34). If for all cluster properties such correction
factors with ~200_1 would exist as “corrections to scaling”, then it would be nearly impossible to get
accurate exponents from finite cluster sizes s, since the corrections would even for moderately large s
not be negligible. These fears turned out to be not justified: Epsilon expansion showed [87] theexponent for s in the leading correction to be ~(6 d) +~ , whereas 1 2o varies as (6— d)2. Thus thetwo exponents are not the same, and the extrapolation of the correction exponent to three dimensions
gave a large value near unity [87].
Thus the concept of nonintegral dimensions and of epsilon expansions has been of practical use forour understanding of three-dimensional systems.
—

—

—

—

2.4. Inequalities

The latest method for the study of large percolation clusters are exact inequalities [88—91]
introduced by Schwartz [88].In particular the results of Kunz and Souillard [90]stimulated further
research. We give here a simplified nonrigorous presentation of these rigorous results.
Assume that for sufficiently large s, at fixed p, the cluster numbers n~decay as log(n,) or
asymptotically [70].This exponent ~ = ~(p) is not a critical exponent in the usual sense since it isdefined for all p, not only close to Pc. Above Pc all available information suggests that the n,(p)decrease with increasing p at fixed s, as one can see already from the concentrations of single sites
and pairs mentioned after eq. (7). For p near unity, only the most compact configurations [47]with thesmallest perimeter tmjn(5) survive in the sum of eq. (7). This minimum perimeter varies as ~for—

D. Stauffer, Scaling theory ofpercolation clusters

19

large s; exact formulas for all s were given in ref. [92]for some two- and three-dimensional lattices.
Eq. (7) gives for p near unity: n, = g,10,,,.p’(l _p)t~~= g,t~j,(1_p)tm~, or log(n,) cx tmjfl
511/d Thus~p~1)i1/d.

(ha)

If for any Pt between Pc and unity the exponent ~ would be larger than 1 1/d it would mean thatcluster numbers n, for such a P’ would decay quicker with size than for p near unity. Thus for
sufficiently large s we would have n,(p1) to be smaller than n, forp near unity; and such an effect was
excluded by our initial statement. Thus we made plausible [90]—~(p>pc)~11/d.

(llb)

In fact the equality holds in expression (1 ib) for a small interval near p = 1 according to ref. [90].
Below Pc we use the numerical result from series [42,93] that the total nwhber ~ g,~of “animals”
increases for large s as s°A’,with e.g. in the triangular lattice A 5.183 and 0 1.00. (See ref.t [94]forcan beinequalities
on
A.)
For
p
close
to
zero
all
animalsineq.
[7]getequal
weight
since
then
(1
—p)
approximated by unity: n,(p —~O)= ~~g,t = p’ ~or s°(pA)’,or
1p’(l—p)(12a)
For general p the same consideration gives an upper bound: n, (p) ~ const s (pA)’ for large s and all
p. A lower bound is obtained also, since n, is always larger than the average number of chains having
the maximum perimeter tmax; for example, in the triangular lattice we have tmax/s = 2 for large s. With
this asymptotic ratio of 2 (which may be replaced by any other constant) we find n~to be larger thanpS(l ...p)tm~ = [p(l —p)2]’. We thus have for large clusters [88]p(1_p)2~<(n,Y’i~~
~pA.

(12b)

Therefore, as long as pA <1, we need ~ = 1 to agree with eq. (12b); otherwise either the upper boundor the lower bound would be violated:~(p<1/A)=l.
(12c)
Furthermore refs. [88,89, 91] derived(g,i)’~”~(a + 1)”~1/a”;

a = t/s(13a)

if t and s go to infinity at constant ratio a. [Proof: Eq. (7) gives for t~= as at arbitrary a: n, =~ ~S~P (I p)t ~ g,~.
[p(1 p)a ]~. This expression cannot increase exponentially with s, and thus weneed (g,~,)’~”p(1p)” ~ 1 for arbitrary a and p. If we use p = 1/(1 + a)or a = (1— p)/p in that inequality
we get (g,~,)”a”/(i+ a)~~ 1, which gives eq. (13a).] Using this result, Reich and Leath [89]derived
—

—

—

(13b)
where PA IS determined by aA = (1 ~PA)/PAand (aA + i)~’/a~= A. In the triangular lattice the above
data for A give a,, = 1.429 and thus P~= 0.412; for the square lattice a,, = 1.023 and P~= 0.494. These
values for P~are already quite close to Pc = 0.50 and 0.59, respectively, suggósting what we will findlater, that ~ = 1 for all p below Pc.
Thus the asymptotic “decay exponent” ~ equals 1 lid above Pc and 1 belOw Pc, at least far awayfrom the critical point. These inequalities do not yet exclude the possibility that there exists a second
and third critical point P2 and p~below and above Pc, respectively, such that C(P <P2) = 1, ~‘(p> p~)=~(P<PA)=l,

—

20

—

D. Stauffer, Scaling theory of percolation clusters

lid, but ~ different and unknown for p between P2 and p~.In that case no drastic changes areallowed in the cluster numbers at these additional critical points, if the “free energy” L n,(p) has only
one critical point Pc where it is not analytic [95]. Our next section will show from Monte Carlo
methods that at least in two dimensions no such complication seems to exist and that the decay
exponent ~ equals unity for all p below Pc and equals I lid for all p above Pc, even close to thecritical point.
—

3. Cluster numbers

This section is the main part of our review and deals with a simple formula, eq. (15), to describe theaverage number of large s-clusters close to the percolation threshold Pc, as a function of s and P Pc.
Numerical tests are presented which confirm roughly that scaling formula. Except if stated otherwise
our results here apply only to large s and small P Pc.
——

3.1. Scaling theory
This subsection suggests a scaling assumption with two free exponents if and T, which are fitted on“experiment”. All the other critical exponents then are derived from these two. In this sense a scaling
theory is phenomenological, i.e. it merely relates different measurable quantities without calculating
any of them directly. In contrast, the “microscopic” numerical techniques of section 2 predict suchmeasurable quantities directly from first principles. The same distinction between phenomenologicaland microscopic approaches applies to thermal phase transitions [10,11] and other problems. After wehave explained exponents we also give a table summarizing the present numerical estimates for themas a function of dimensionality.
3.1.1. The basic assumption
The critical behavior near Pc is characterized by the fact that the typical cluster size becomes verylarge near the phase transition and diverges at Pc; otherwise the typical cluster radius ~ of eq. (6e)would not diverge. One may define the “typical cluster size” Sg as that size which gives the main
contribution to the singular parts of L s2n~,~, s3n5 or of any other sum appearing in eq. (6). (Thistypical cluster size is not identicalto anotherwhich
the avoid
literatureis unfortunately
called2n,/~.
~ n~,a expression
wording which
weinwillhere.)
We assume thatall
the meanclustergive
size:theS same~ s typical size s~apart from numerical constants, i.e. that the different s~
thesedefinitions
all diverge with the same power of I~ pCI. This exponent of p Pc is sometimes called the gap
exponent; we denote it here by 1/if. Thus our basic postulate is:——

We assume that the critical behavior of percolation is dominated by clusters of size s~or~
where differently defined typical cluster sizes all diverge with the same exponent.
(14a)
In short, we assume an essentially unique typical cluster size. “Critical behavior” in assumption (14a)refers to the singular parts, as defined in section 1.2. For example, s~is the size dominating in~‘Is ]sing, but not in ~. s n
5 singularities come from typical clusters, analytic background terms
arise from much smaller clusters.
How can we put eq. (14a) into a quantitative form? All singular cluster properties must depend
—

S

Sg

D. Stauffer, Scaling theory of percolation clusters

21

mainly on the single ratio s/sg since the typical cluster size was assumed to be unique. If instead theywould depend on two variables s/sg, and s/se, we would have more than one typical cluster size s~,incontradiction to eq. (l4a). But the assumption n~(p)= f(slsg) would be too simple, since then at P = Pc,
where s~= ~, the cluster numbers would be constant, in contradiction to fig. 3. It seems plausible, justas we expressed the cluster size by the ratio S/Sg, to calculate also the cluster numbers through theratio n5In~and to assume that this ratio depends only on the ratio s/sg:
n5fn~= F(s/s~).

(14b)

Since the typical cluster size is defined only up to numerical factors, this assumption (14b) is not verypractical (except for one dimension: appendix 2) and we replace the ratio n5/n5~by ns/ns(pc). (The final
result (15) is the same.) Thus:
We assume that the ratio v3(p) = n5(p)/n,(p~)and similar ratios of other cluster properties are afunction of the ratio s/se only:
v5(p) = F(s/s~).
(14c)
Equations (l4b, c) explain why theories of this kind are called scaling theories: If we plot n~versus sfor different p Pc and if we scale the cluster size s by dividing through and the cluster numbers
n,(p) by dividing through n,(pc) or ~ then in these scaled variables the pkts are independent of
P Pc
We have already seen in fig. 3 that at p = Pc the cluster numbers ns(pc) decay with a simple power
law: n5(p~)or s~. And now we understand why: Any more complicated decay law, like ns(pc) or
s~exp(—s/so), would mean that at Pc there exists a finite typical cluster size s0. Since s~(p= Pc) is
infinite that result would mean that more than one typical cluster size exists, in contradiction to ourbasic assumption. Thus eq. (14c) reads: n~(p)or s~F(s/sg).Since s/sg or I(~ pc)soIhbo, we may summarize our assumption in its final form:
Tf(z)with z (p Pc)5~and f(0)or= ns(p)1. or s(15)
The normalization f(0) = 1 comes from the trivial requirement VsQ~c) = 1. This assumption (15) is thebasis of our interpretation of cluster numbers; we will determine the exponents g and r and thescaling function f(z) from microscopic methods later, since they are not predicted by the present
phenomenological scaling theory.
Eq. (15) was first mentioned by Stauffer [67]as a generalization of the Fisher droplet model [34];for thermal phase transitions analogous generalizations were made earlier [96]~(See also refs. [18,97,
98] for related assumptions.) This Fisher model [34]simply assumes n
5 cc ~ exp(—const z) for p
above Pc, which is clearly a special case of eq. (15). Although this Fisher formula finally turned out tobe quantitatively inaccurate, as we will see later in fig. 7, it gives a rather goo4 approximation for the“ferromagnetic” side of the phase transition and played an important role in the development of thescaling theory for percolation clusters. Many of the qualitative discussions of this section and of
appendix 1 can be followed easier if the reader works with this simple Fisher formula instead of withthe general scaling assumption (15). We may also2), ~ regard= ~, i =eq. (15) as a geneitalization of the Bethe
latticeresult[30]where
n5
cc
~5I2
exp(—const.
z[For finite s the cluster numbers n, are a polynomial in p, eq. (7), without any singularities at
p = Pc. Thus we expect also the limit function f(z) = n,/n,(pc) to remain regu’ar at p = Pc, i.e. z = 0.
The scaling function 1(z) is therefore assumed to be analytic in its argument z f~rall z including z = 0.
For large z, the scaling function f must decay sufficiently rapidly to make all the sums in eq. (6)converge.]
—

Sg

22

D. Stouffer, Scaling theory of percolation clusters

3.1.2. Relation to critical exponents
Now we relate the cluster exponents if and T to the other critical exponents a, /3, y, ~ defined ineq. (6). In order to evaluate the singular part of any moment ~, s”n~qualitatively, i.e. in order to get its
critical exponent correctly, we may replace in the sum each factor s by the typical size s~and thecluster numbers n~by the numbers of typical clusters n5~cc sTf(const) or s~.Moreover the summation over all cluster sizes s gives an additional factor sc: [~s S k~5]sing cc 5~+k~ Since sE or
Pc~~
by definition, we thus have [18]—[~skn~(p)]singor p —

pcI(Tl_~~~.

(16)

A more rigorous derivation of this exponent, together with evaluations of the proportionality factor, isgiven in appendix 1. A reader who mistrusts our simple approximation above and does not want to go
through the details of that appendix may check eq. (16) by simply using the Fisher formula,or s~ exp(—const~z), and by evaluating the higher moments like k = 2, 3,. . . only. The integralsthen can be expressed by gamma functions [99].
Eq. (16) gives us immediately the various exponents of eq. (6); we merely take k = 0 for the numberof clusters (eq. (6a)), k = 1 for the percolation probability (eq. (6b)), k = 2 for the susceptibility(eq. (6c)) to get a, /3, y as [34]:(17a)2 a = (T 1)/if,/3 = (T 2)/if,—y = (TRight at p = Pc, the sum in eq. (6d) defining the exponent ~ arises mainly from cluster sizes s near h/h.
Thus the nonanalytic part of ~, .
varies as h’h~h~= h~2,as appendix 1 again will show
in detail; and we therefore have3)/if.

————

fls(pc)C~

1/6—T2.

(17b)

We may solve these four equations to get the cluster exponents as a function of the other exponents
and to derive two relations between the latter exponents:
T=2+1/eS;o~=1//3~=1/(y+f3)(18a)
2—a y+2/3/3(~+1)[dv].
(18b)
The last relation, where the dimensionality d enters, will be derived later in eq. (37c). These famous
scaling laws (18b) are known since many years, also for other phase transitions [10,11, 30]. If wewould not have gotten them, we would have to change our scaling assumption (15), not the scaling
laws. Again eq. (18) comes as no surprise since the same relations were known from the Fisher model[34];in fact their first derivation for thermal phase transitions by Essam and Fisher [31]used thiscluster model.
In this sense we fulfilled our promise to use percolation as a simple introduction to scaling laws: By
a plausible assumption (15) on the cluster numbers we could derive the desired relations between
critical exponents, eq. (18). Of course, it is no great revelation that we found four relations between
six exponents, when we made an assumption which contains only two exponents as free parameters.
Our assumption (15) is completely analogous to experience made with other phase transitions. Thatexperience has shown: Any “singular” function g depending on two “critical” variables g(x, y) can,for small x and y, be written in general in the form g(x, y) = f G(y/x”) with two free exponents b
and c. Our scaling assumption (15) is merely a special case of this general “scaling homogeneity”
assumption [100].(It is not easy to write down an explicit expression for g(x, y) which does not reduceD. Stauffer, Scaling theory of percolation clusters

23

to that simple homogeneous form for small x and small y.) For our percolation clusters we simply
identified x with 1/s and y with p ~Pc, and gave special names to the two exponents to get
TG((p ~Pc)5~) in agreement with eq. (15). Percolation clusters follow what can be learned from= sother phase transitions and have no intention to learn new tricks. For some readers that might makeassumption (15) more likable since it now has the form: “Percolation clusterS are lazy and invent no
new ways to deal with critical phenomena”. For the reader who is also lazy and does not want to readabout equation of state or universality we recommend to go directly to tables 2 and 3 at the end ofthis
subsection 3.1.
3.1.3. Dilute ferromagnets and ghost fields
The equation of state of the dilute low-temperature ferromagnets mentioned in section 1.1 can also
be brought into the scaling-homogeneity form mentioned above as customary for critical phenomena.
The mathematics is simplified if we look at the susceptibility x aM!
13H since~then the singular part is
identical with the leading divergence, and the analytic 6)withbackground
does not disturb
us. At thermal
the abbreviatiØn
h —p~H/kBTfor
the
phase
transitions
one
has
x
or
ITO—
T~X(hIIT~—
TI’~reduced magnetic field. The scaling function X must be a symmetric function of its argument, leadingto ~(—h)= ~(h), since the behavior of a magnet is independent of the directiOn which we choose forthe coordinate axes. For the dilute magnet, eq. (5), we find by differentiatiOn, using d(tanh y)/dy =cosh2(y) and z = (p —p~)s~

x pkBT/~=p= ~aM/ah = ~ s2n2(sh) cc

5(p)coshIP _pc!(T3~~

J

J IzI

J

52_Tf[(p~ST)/cTf(z) cosh2(lzJ”°h/Ip pcj~k~)dzf

I~—p~L” IzV~
f(z) cosh2(Izl~~
h/Ip —p~I~)
dz

cc

(19a)

with the z-integration running from 0 to +~ for p > Pc and from 0 to
for p <Pc. Thus our
susceptibility obeys the simple homogeneity form postulated above, but with p Pc replacing Tc—~—

x Lo —p~I~’X(h/Ip .p~~B0).

—

(19b)

=

Eq. (19) gives an explicit scaling expression, in terms of a cluster scaling function f to be determinedlater, for the susceptibility; it has the full symmetry, homogeneity and analytkity properties required
generally near critical points. For thermal phase transitions that problem is much more complicated;
the simple Fisher model, for example, violates the symmetry requirement [34]. No explicit formulas
have to our knowledge been proposed for critical phenomena in other cases, ~~vhich
do fulfill all theserequirements. As promised, percolation is a particularly simple phase transition; it merely requires
substitution of integration variables as mathematical tool so far.
For the magnetization M in eq. (5) and the free energy F = fM dH analogous scaling-homogeneous expressions can be derived from eq. (15). The same is true for the generating function or
ghost-field free energy [~,n5(,p)~_lt5]~j~g
and similar expressions for the highe~moments. In fact, theresult—n, e_h5]
5

.

sIng

= I~~

P(hfIp —p~I~)

(20)

24

D. Stauffer. Scaling theory of percolation clusters

has been postulated by Essam and Gwilym [30]in the first scaling theory of percolation, long before
cluster scaling was suggested [67].Nakanishi and Stanley [23]recently confirmed that assumption
numerically by Monte Carlo computer simulations. It is rather obvious that a scaling assumption (15)with two free exponents for the cluster numbers n5 leads to a two-exponent scaling expression for the
moments or generating functions, as in eq. (20). But the reverse is not true, see appendix 1: One caninvent expressions for the cluster numbers n~with three free exponents, which lead to only two free
exponents in the scaling expression for these sums over cluster numbers. Thus the assumption (20) of
Essam and Gwilym [30]is a consequence of our assumption (15) but is not equivalent to it; and moredata than those of Nakanishi and Stanley [23]are needed to confirm eq. (15).
Therefore eq. (15) has to be tested directly since scaling laws alone do not prove it. In fact, also
alternatives to eq. (15) have been proposed which do not employ just two free exponents but more or
less. In the first scaling theory for cluster numbers n5(~p), Domb [101]suggested to take ~ = ~and touse only one free exponent r. And later Leath [73]made an assumption which corresponds in many
applications to a scaling ansatz with three free exponents r, 4, ~!i. But later work [54,71, 83, 102]
showed if <~ and q5 = ~‘i, leaving us again with two free exponents. Very recently Kunz and Payandeh[79]showed how renormalization group makes assumption (15) plausible. Thus at the time of thiswriting eq. (15) seems to be the only viable scaling assumption still floating around, and weconcentrate on that assumption when we present numerical tests in subsection 3.2.
3.1.4. The universality concept
The “universality” concept [11,103] for thermal critical phenomena is a semi-empiricalclassification of different systems into few universality classes. It asserts that critical exponents andother qualitative properties like the shape of the scaling functions are independent of details like range
of interaction or structure of the lattice. Only the dimension is important enough to change exponents.
A precursor of this universality assumption is the law of corresponding states for fluids. Universality
in this naive sense turns out to be valid in most but not all cases. Thus for the majority of thermal
phase transitions one can predict rather reliably which different systems belong to the same
universality class. Then one can predict critical exponents without having to solve in detail the criticalbehavior of these systems. This universality concept thus saves a lot of work if one can trust it;
roughly speaking all three-dimensional systems then have the same exponents, etc.
For percolation, we postulate similarly [50]that exponents and the shape of the scaling function
f(z) in eq. (15) do not change if we change “minor” details like the lattice structure; only thedimensionality is important since, after all, it appears explicitely in the scaling laws (18). Morequantitatively we may generalize the scaling assumption (15) for one lattice, n5 cc s~f[(p —p~)s°i,
to auniversality assumption for all lattices of the same dimensionality d [103]by introducing latticedependent scale factors q0 and q1 besides the lattice-independent exponents if and r. Also we denote a
lattice-independent scaling function by f~,where u stands for universal, and require
n5 =q0s~f~[(qi(p—p~)s°]; f~(0)°~1.
(21)
This universality assumption (21) means that for a given p the shape of the size distribution n5 is
the same for different lattices of the same dimensionality d; only the two scales for the n5-axis and thes-axis are different. For example, if in the triangular lattice for large clusters the maximum value ofn,(p) (as a function of p. at fixed s) is five times larger than n5(p~),then also in the square lattice themaximum is five times higher than the critical value for the cluster numbers. But for three dimensions
2n
this ratio can be lower. Similarly, if one calculates the ratio of “susceptibilities” ~ s 5(p <D. Stauffer, Scaling theory of percolation clusters

25

pJ/~5s2n5(p > p~)for equal distances ~p p~jfrom the critical point, then scaling requires the ratio to
be independent of p p~for one lattice structure, and universality requires it to be the same fordifferent lattices of the same dimensionality d. (How this ratio depends on d is not yet known ingeneral [26].)More quantitatively, the ratio can be expressed by integrals involving only the universal
1~~~where the lattice dependent factors q0 and q1 have cancelled out:
—

—

2n
2n
~ s 5(p <~~)/~s 5(p > p~)=fy”’f~(—y)dy/f y~f~(+y)dy

with y = q1(p pc)sU, as can be checked easily with the methods used already in eq. (19). (One may
also postulate relations between the coherence length ~ and the “free energy” >~n5 analogous tothermal phase transitions [104];but little is known on that subject, which would be anyhow outside
the scope of the present review.)
Marro [105]made several numerical tests involving similar ratios and found them indeed universal
as desired. The critical exponents are universal, within small error bars, in two and three dimensions
when various lattices are compared [54].With larger error bars, exponents seem unchanged [24]ifclusters in percolation are defined not just as nearest-neighbor connected groups but if also longer
distances were allowed to connect the sites; but from now on we will again deal only withnearest-neighbor connected clusters. For the cluster size distributions, the universality assumption(21) was confirmed quite directly in both two and three dimensions for selected lattices [26,71]. Thus
it seems legitimate to regard critical exponents as a function of dimens~onalityd; our table 2,following Stanley [106]and more recent sources [107—110,
7, 26, 60, 71, 93], compiles averaged
numerical estimates for many exponents as a function of dimensionality d.
We see that the scaling laws (18) are confirmed reasonably well in two and three dimensions, just as
they are known to be valid in thermal problems. In the three-dimensional Ising magnet near its Curie
temperature, there is a problem with the latest scaling law dv = 2— a; but according to ref. [111]thetwo error bars overlap. Similarly for two-dimensional percolation the estimate 2— a = 2.668 ±0.004 of—

Tab’e 2
Numerical estimates of directly determined critical exponents. If one believes inscaling laws, eq. (18), one can fill in the empty spaces. The exponents U and ~sforanimals and conductivity are defined later, eqs. (27—49). Our values are averages overdifferent methods, if available; the error bar is seldomly more than one unit of the lastdecimal. Perhaps v(d = 2) = 1.355 is exact, eq. (9). References are given in Stanley’stable [106];see also refs. [7, 17, 60, 107]. Data for the exponents 0, p. 0’ and r fromrefs. [26, 71, 93, 108, 109]. One-dimensional percolation is discussed in appendix 1.
Infinite dimensionality corresponds to “classical” results [27—30,114, 149], they are
expected to be valid already above six dimensions.
d

a

$

y

S

v

1

1
—0.67

1
2.43
1.7

~
18
5

1
1.35

3

0
0.14
0.4

0.84

0.4~

2.1,

4

0.5

1.5

3.~

5

0.7

1.2

6

1.0

1.06

I

1.0
I

2
2

2

7
~

—1

~

0

p

1

2

0.39

2.0

0
1.00
1.5

1.2

0.7

1.9

2.4

2.~

0.6

2.2

2.7

2

0.5

2.4

3.0

iT

—

1.7

2.4

1

1

1

1

3

26

D. Stouffer, Scaling theory of percolation clusters

ref. [56]disagrees with dv if ii = 1.355 is exact, eq. (9). The estimate for 2—a also does not agree wellwith other series estimates [107,110] y = 2.425 ±0.005 and /3 = 0.139±0.003and the scaling law
2— a = y +2/3. In the other cases no such problems occur at present.
One very important non-universal parameter, which moreover is not predicted by any of the above
scaling theories, is the position p~of the critical point. Scaling and universality tell us something about
what is happening near p~,but tell us nothing about where Pc is. Table 3 lists numerical estimates andexact results for various lattices and various dimensions; no reliable connection exists at present
between p~and the critical exponents. In the triangular “site” percolation problem and the square“bond” percolation problem we have p~= ~, presumably exactly [24];moreover Pc = 2~sin(~ih8)=
0.34729 for triangular bond percolation and p~.= 1 — 2~sin(~r/18)= 0.65271 for bond percolation in thehoneycomb lattice. Clearly universality cannot assert that p~is the same for all lattices.
Returning to the universal (?) exponents, we unfortunately have to mention the results of Klein etal. [112]that in one dimension the percolation exponents depend on the range of interaction, in
contrast to higher dimensions [24].This clear violation of law and order is discussed in detail inappendix 2; clearly the naive universality picture as suggested here is not valid in all cases. The
special status of one-dimensional percolation is also evident from table 2, where d = I does not seemto follow for some exponents the trends evident from d = 2, 3,~. . [133]even when only nearestneighbor connections are used. On the other hand the behavior for larger dimensionalities is rathersmooth and agrees quite well with the epsilon expansion, eq. (10), even though only the first two termsof an expansion in
6— d are used. This agreement is seen clearer in fig. .5 where /3 and y areplotted for one to seven dimensions.
We conclude from the information presented here that the scaling laws for the critical exponents
like /3 are confirmed rather well, that these scaling laws can be explained by a simple assumption on
Table 3
Numerical estimates (three digits) and(presumably) exact results (fourdigits) for the percolation threshold
p~. TR = triangular, SQ = square,
HC = honeycomb, D = diamond, SC =simple cubic, BCC = body-centeredcubic, FCC = face-centered cubic.
Results for higher dimensions refer tohypercubic lattices. From refs. [7, 26,
44, 45, 54, 55, 60, 107].
Lattice

site

bond

HC

0.698

SQ

0.593

0.65270.5000TRD

0.5000
0.428

SC

0.311

0.249

BCC
FCC
d=4
d=5
d=6
d= 7
d—6=

0.245
0.198

0.178,
0.199

0.197

0.161

0.3473

0.388

0.141
0.118
0.108
0.094
0.089
0.078
p~=1/(2d— 1)

D. Stauffer, Scaling theory of percolation clusters

27

Fig. 5. Plot of the two independent exponents $ and y as a function of dimension d. The curves represent eq. (10), the epsilon expansion. Datafor y are directly from table 2; datafor $ also employ the scaling laws $ = ~(8 — 1)= 1(dv — y) and represent average estimates: $ = 0.41,0.58,0.76,
1.00 in 3,4,5,6 dimensions. In two dimensions, $ = 0.139±0.003[110].

the cluster numbers, and that we have to test this scaling assumption directly. This is done in thefollowing subsection.
3.2. Tests of scaling
Only for two and three dimensions are cluster statistics known to us which have been analyzed interms of scaling, eq. (15); the polynomials in refs. [45,107] are still waiting fOr a “lover” to study n5
for general d. And Monte Carlo work in more than three dimensions has not yet been done on n5. (In
5.99 dimensions, epsilon expansion results of Stephen [114]were consistent with eq. (15).) We restrict
ourselves here mainly to recent Monte Carlo work of high accuracy by St~lland Domb [62,115],
Leath and Reich [71,72] and Hoshen et al. [26],each based on many millions of Monte Carlo step foreach concentration. We mentioned already before that the Monte Carlo Work of Nakanishi andStanley [23] for the “equation of state” gives an indirect confirmation of the cluster scaling
assumption (15).
The various regions to be investigated can be seen rather elementary if we look at the number n2(p)of pairs in the square lattice: Each pair consists of two occupied sites sutrounded by six2(hemptyp)6neighbors,
it canhas
be aoriented
eitheror vertically.
we have
n2 =the2p position
exactly.
Thisandfunctionmaximum
at p horizontally= 0.25, far below
p~= 0.593;Thusfor larger
clustersof the maximum shifts closer to p~,and the peak becomes more narrow. Thus generally we may lookat what happens at p~,above p~,near the maximum below p~,and asymptotically in the wings
relatively far away from the maximum where n
5 becomes extremely small. The latter questions will
also be discussed outside the scaling region, i.e. with p not very close to p~.
—

3.2.1. At the percolation threshold
Right at Pc the scaling assumption (15) asserts n, s~.We need T > 2 since the total number ofoccupied sites, ~ s n5 = p~,from eq. (4), must remain finite even at the percolation threshold. Figure
3 in the introduction already showed two-dimensional results, right at the exact p~= ~of the triangularlattice, giving such a simple power law. An analogous plot for three dimension~is given in fig. 6. These
two computer simulations used the largest lattice known to us for Monte Carlo work, 4000 x 4000 sites
in two and 400 x 400 x 400 in three dimensions. The solid line in figs. 3 and 6 ~snot the best fit on theMonte Carlo data but the prediction from series expansions, n5 cc ~2II8 with both the exponent andthe proportionality factor calculated from refs. [19,55]. (If n5(p~)varies as s~ for large s,

28

D. Stouffer. Scaling theory ofpercolation clusters

10_15

0

I

I

102

101~106

Fig. 6. Monte Carlo cluster numbers [261at p = p. = 0.311 in the simple cubic lattice, together with the series prediction [55].

then the relation r = 2 + ho is exact and does not require the validity of eq. (15). Thus figs.
3 and 6 constitute only a weak test of the specific scaling assumption used here.) The agreement ofMonte Carlo and series results is excellent on the scale of figs. 3 and 6.
Closer inspection, however, shows problems: The best fit for r from large s in figs. 3 and 6 isT(d = 2) = 2.02 and r(d = 3) = 2.135, significantly lower than the series predictions [19,55] r =2.055 ±0.002 and 2.2 ±0.03, respectively. (Monte Carlo data in ref. [71]give r = 1.97 ±0.05 in thetriangular lattice at Pc, but T = 2.05 ±0.02 from scaling and all p.) There are two reasons [26]for these
deviations, which we will discuss now: Neither the cluster size nor the lattice size are large enough forthe asymptotic scaling behavior to be exact in the computer experiment; the true exponents should
show up only for s ~ in an extremely large lattice. Present computer technology unfortunately does
not allow an infinite number of operations with an infinite computer memory during a finite time atsmall costs; thus Monte Carlo studies are always experiments with both systematic and statisticalerrors.
More precisely, already for the exact cluster numbers n5 for small s from series expansions [46]
one can see that at PC the measured n~were below the extrapolated cluster numbers represented by
the solid lines in figs. 3 and 6. If we assume r 2.05, as required in two dimensions, then a muchbetter fit over nearly the whole range of s was obtained [26]for the triangular lattice by—~n, = 0.03(1 1.2(22)
where a first correction term to the simple asymptotic power law is used. This correction-to-scaling
exponent 2/3 from Monte Carlo data agrees roughly with the analogous exponent 3/4 found forbasically the same quantity from series expansions [19]and is consistent with the order-of-magnitude
estimated [87] from the epsilon expansion [87]. If eq. (22) is correct, then the apparent sloped(log n5)/d(log s) in a log—log plot as in figs. 3 and 6 deviates from the true (asymptotic) slope r
as r. = r — 2.2~~.
and this deviation is, for s 102, about as large as the difference between theMonte Carlo and the series value for r. Quite generally, it is difficult to get from Monte Carlo method
reliable values for the second decimal after the point in a critical exponent; and series extrapolations
usually give more accurate exponents.
The influence of the finite lattice size can already be observed for very small clusters. E.g. in
ref. [26]for isolated sites (s = 1) the actually observed cluster numbers in the Monte Carlo experimentwere found to be larger by about 0.35 percent than the exact cluster numbers for infinite lattices. This~_2.O5—~_2~l3)

—

~213,

—~

D. Stauffer, Scaling theory of percolation clusters

29

deviation increases with increasing cluster size s and is responsible for the deviations observed in
fig. 3 for large s: the boundaries of the system break up some very large clusters into smaller clusters
and isolated sites. The use of periodic boundary conditions (section 2.2.1) reduces these “finite size”
effects [69,62, 115]; in ref. [26]periodic boundary conditions made the numbers of small clusters agree
within statistical errors with the exact results and shifted the fitted exponent r upwards to 2.16, closer
to the series prediction 2.2, inspite of the fact that the system used was smaller than the one used for
fig. 6. (In Leath’s method [71—73] there is no disturbing influence from the boundaries.) Thus we tookr = 2.15 as the direct Monte Carlo estimate in table 2, although the true value is presumably slightly
higher. Clearly a direct evaluation of r = 2+ 1/0 from Monte Carlo cluster numbers at p~,is not a very
accurate way to determine the exponent 5; and series methods [19,55] work better here. We now turnto the region p Pc where series methods gave only rather crude extrapolations [51]for the cluster
numbers and their scaling function; here Monte Carlo methods can be mOre reliable than series
extrapolations. (Also, in the following discussion we are satisfied with an accuracy of about 10
percent; all the above problems concern only the fine details of the second decimal.)
3.2.2. Above the percolation threshold
The cluster numbers above PC decay monotonically with size. Figure 7 shpws logarithmically thevariation of the Monte Carlo ratio v5(p) = nS(p)!flS(pC) with the scaling variable z = (p —p~)s~
in thetriangular lattice. Here the exponent a’ was taken from the above results as a’ = l!(y + /3) = 0.39 [54].
If assumption (15) is correct then the values for different p PC should all follow the same curve sincez is used as a variable here. This is the basic result of scaling: By multiplyiflg the original variables
p p~and n~with suitable powers of s, or by similar rescaling of the axes, the original data are shifted
such that different data (e.g. for different s) follow the same curve. We see that in fig. 7 different
symbols, representing different p — Pc in this case, follow the same curve within the “experimental”
scattering. This figure thus confirms our basic assumption (15) for the cluster numbers. The curvethrough the points in fig. 7 is the extrapolation from series results [51]and agrees remarkably well with
the Monte Carlo data. For three dimensions, similar confirmation of scaling by Monte Carlo results
was obtained, too [26].
The tangent to the origin in fig. 7 arises from additional more accurate cluster numbers obtained
very close to PC~Taken together, fig. 7 shows convincingly that the data do nc~tfollow a straight line;
—

—

0

Q~

I

I

0.2

Q3

0.4

05

Fig. 7. Scaling plot above p~.Variation with z = (p — p~)s~of the ratio v, = n,(p)In,(p~)on a logarithmic scale. Assumption (15) requires different
symbols to follow the same curve. Monte Carlo data from 4000 x 4000 lattice. From Hoshen et al. [26).

30

D. Stouffer. Scaling theory of percolation clusters

0,J,hlIII1.&_________

P,

I

I

I

0.001

1O~

102
p...pSC

Fig. 8. Variation with p of the fiaction P,. of occupied sites
belonging to the “infinite” network in a 4000x 4000 triangularlattice. p is shifted from p. to Pc + 0.00085 to offset some of theboundary effects in the Monte Carlo calculation. From ref. [26].

I

100

1000
10000 SFig.
9.
Variation
ofP.~..xwith cluster size s in the triangular lattice[261.The cluster number n,(p) reaches at fixed s a maximum at= p,,,.~below p~.
The fitted line has slope —0.40 in this log—logplot; scaling, eq. (15), requires the slope to be —0.39. Data for verylarge s are unreliable because of the finite size of the 4000 X 4000
triangular lattice.

pthe upward curvature proves that log v. is not simply proportional to —z = — (p — Pc)5. Therefore thesimple Fisher droplet model above PC, with n
5 cc s_re_z Const is not exact since it would give a straightline in fig. 7.2aInstead,astraight
line
isobtained
instead of z. Thus for not too rather
small wellz we[26,62,
find n115] if the data for v5 are plottedagainst z~
5 or ~ exp(—const~\/s) in two
dimensions, for the triangular and square lattices [26,62,115] even when some correlations between
occupied sites are taken into account [115](see section 6.2.3).
If we define a decay exponent [70]~= ~(p) throughlog n5 cc—(s —~xat fixed p)

~

(23)

then these data suggest 2~~~= already~ in twofordimensions.
Thus
the scaling
f(z)notin merely
eq. (15)forseems
rather small
z values
abovefunction
0.1, andz ~ to
1.
decay
as
log
f
cc
zU
Therefore the simple formula n, cc s~ exp(—const~Vs) fits most data quite accurately above thepercolation threshold; according to assumption (15) this constant in the exponent vanishes as(p p~)~2
at the threshold [115].
The percolation probability ~ cc (p P~)~is particularly difficult to measure by Monte Carloexperiment, since in two dimensions /3 = 0.14 [110]is so small. In a finite lattice with free boundaries
one may approximate P,~by the fraction of occupied sites belonging to the largest cluster; usually in a
Monte Carlo experiment slightly above PC the largest cluster is clearly larger than all the otherclusters. The P~.determined in this way [26]obeys with surprising accuracy the desired power law;
the slope of the line fitted in fig. 8 through the data is 0.133, very close to the series prediction of/3 = 0.139±0003[110].Also the proportionality factor agrees well with the series prediction [26,54].
In three dimensions the exponent for the percolation probability is /3 0.4 [7,60, 110].
Overall, above p~,no contradictions were observed between different Monte Carlo simulations,series extrapolations, and scaling, assumptions; moreover we can fit the data with log v
5 cc — \/s, i.e.
with ~= ~over most of the range for d = 2.
———

3.2.3. Below the percolation threshold
Below Pc the cluster numbers n5(p) as a function
p at Pmax
fixed=s1/4.
show
a maximum (15)
at P is= Pmax,as2(1of p)6,
If assumption
correct,
we discussed before in the simple case of n5 = 2p—

D. Stauffer, Scaling theory of percolation clusters

31

this maximum corresponds to a maximum in the scaling function f(z) at some negative argument
Z = Zmax = (Prnax — p~)5G~(This quantity Zmax is fixed for a given lattice, but depends on the lattice type.)
Fig. 9 shows [26]how Prnax approaches p. for large cluster sizes: Pmax p~cc s°,with an exponent
a 0.40 fitted on these data. Another scaling test [26]uses d(n5)/dp and gi~’esa’ 0.38. Leath andReich [71],also from Monte Carlo studies, conclude a’ = 0.394 ±0.003 for triar4gular and square lattice.
Thus our table 2 gives a’ = 0.39 as the directly determined cluster exponent~This value happens toagree with a = 1/(y + /3) = 0.390 ±0.001 as predicted by scaling theory and t1~elatest series estimatesfor y [1071and 13 [110].In three dimensions [261direct Monte Carlo detern~inationsof o• were lesssuccessful; values between 0.4 and 0.5 agreed with the data, leading to our esdmate 0.45 in table 2; thescaling prediction is a’ 0.48 from series expansions [54,110].
Figure 10 shows that the scaling assumption is confirmed by two-dimensional Monte Carlo data
[26,721: As required by eq. (15), different symbols representing different p Pc follow the same curve,
just as they did in fig. 7. The short curve near the origin symbolizes numerOus additional data [261measured there for smaller values of the scaling variable z; they also agree with scaling, eq. (15).
Again these data roughly agree with extrapolations [51]from series expansior~s.
Figure 10 gives a maximum of v5(p) = f(z) at z = Zmax = 0.41 ±0.03, with a height fmax = f(Zmax) =4.9; universality asserts this fmax to be the same for all two-dimensional’lattices. If the scaling
assumption (15) is correct then this maximum in n5(p) or v5(p) as a function of p for fixed s mustcorrespond to a maximum in v5(p) or fn5(p) as a function of s at fixed p below Pc, since v5(p) equalsf[(p p~)s°].
The position of this maximum Smax(P) is at Smax (Zmax/(P p~))~”.
Indeed, the insert in
fig. 10 shows this maximum quite clearly, using data of Reich and Leath [72]for the triangular lattice.
Similarly for the square lattice Stoll and Domb [115]also find such a maximum in v5 versus s,compatible with Zmax —0.4. The height of the maximum, i.e. the value of ~found from the dataof Reich and Leath [72]is compatible with the above estimate of about 4.9, a~indicated by the error
bars in the insert of fig. 10. Stoll and Domb [115]quote fmax = 5.4 taken from their data farther awayfrom p~.An earlier extrapolation of series results [50]gave fmax = 4.5 for triangular, square, andhoneycomb lattices. Thus we conclude
—

—

—

—

—

—

I

•

II

I,

0

1ILO

5~I

I

I

ye

+

15~

I..,
U1

I
1 Pr

I

.

047

-

2 -

•

..(145e()~455+0.46

i°I’~.2~
—2~

x0.47

.~“
I

p

vO.465

I

I

I

I

I

0.2
0.6
1.0
08 s of the ratio v,(p) below p
Fig. 10. Variation with zI”~= (p,, — p)
5. analogous to fig. 7 above p5, in the triangular lattice [26].Scaling requires
different symbols to follow the same curve. The small dots for p = 0.45 are from Reich and Leath [72].The in~ertshows the variation of v,(p)with a at fixed p. with data from ref. [72],giving a maximum roughly where predicted by scaling [261and fig. 9. (See also ref. [115].)

32

D. Stauffer, Scaling theory ofpercolation clusters

fmax5

(d2)

as the combined estimate in two dimensions.
Leath and Reich [71] investigated scaling theory by Monte Carlo simulations of the triangular andsquare lattice and found eq. (15) to be valid for clusters containing more than 85 sites. For smallerclusters systematic deviations were observed which are larger than but in the same order-ofmagnitude as those given here in eq. (22). It would be interesting to see in future research whether
these deviations also follow a scaling form [116]like v5(p) = f(z) . (1 + ~_0.7f1(z) + ~ as a generalization of eq. (22). But such an analysis would require very accurate Monte Carlo data. (Refs. [71,73]
also determined the cluster perimeter distribution which in turn was used to get more information oncluster numbers. We refer to section 4.1 and ref. [71]for details.)
For large values of the scaling variable z = (p PC)S~ the cluster numbers decay rapidly withincreasing —z; this wing on the left side of the maximum determines the exponent ~of eq. (23). The
straight line fitted in fig. 10 through the data suggests log v5 cc (—z)”’~= (p, p)fl&s cc s. Thus the
exponent ~is about unity below Pc and therefore different from its value~in twodimensions above Pc. (The
first indication of suchan asymmetry came from Bakri [70].)Similarly, Müiler-Krumbhaar and Stoll [37]concluded ~= 1.1 ±0.1 in the square lattice at p = 0.50, rather far below Pc = 0.593. Leath and Reich[71]found that the choice ~ = 1 fits their Monte Carlo data better than the choice ~= 2a = 0.8. And
most recently, Kinzel [80]found ~= 1 over sixty decades in n5, using real space renormahization in thetriangular lattice. Thus for sufficiently large values of IzI we have in two dimensions rather reliably
established for p below p~:
—

—

—

—

—

~(p>p~)=~~~(p<p~)=1.
(24)
When we combine data for the cluster numbers above, at and below p~in one plot [26,51] they
follow roughly a Gaussian curve, as first 2].
suggested by Leath [73]:(25)v5(p) = f(z) cc exp[—const(z Zma~)
Of course, eqs. (24) and (25) contradict each other since eq. (25) gives ~= 2cr 0.8, violating the exact
inequality ~ 1— lId of Kunz and Souillard [90] above p~,eq. (lib). Thus the Gaussian fit is areasonable approximation but not an exact solution for the scaling function f(z). It may replace theFisher model, f(z) = exp(—const z) above Pc, as a simple numerical approximation. Just as the Fisher
model it does not work asymptotically for large z, but in contrast to the Fisher model it can be used on
both sides of the phase transition.
In three dimensions, Monte Carlo data [26]for the simple cubic and the body centered cubic
lattices [26]could be fitted well by a Gaussian approximation as in eq. (25), with ZmaxIPC = 0.8. An
example is shown in fig. 11, where again data for different p Pc follow the same curve since Vs ~5plotted logarithmically against z; thus the scaling assumption (15) again is confirmed. The solid line
there is the parabola corresponding to the Gaussian approximation (25) and fits very well; ref. [26]does not determine the asymptotic decay exponent ~ The maximum value of the scaling function was
found to be [261———fmaxl.6(d3)

in these two lattices. In one dimension we will see in appendix 2 that fmax = oo. For infinite
dimensionality we expect the Bethe lattice solution frn&x = 1 [30]to be valid [45,84, 107, 114, 117].
Thus the above estimates fmax(d = 2) = 5 and fmax(d = 3) = 1.6 fit nicely into the general trend as a

33

D. Stouffer, Scaling theory of percolation clusters
1

lnv

1

I

I

5

+1

Fig. II. Scaling plot in three dimensions on a simple cubic 100 x 100 x 100 lattice. Data for different p follow the same curve, as required byeq. (15). The parabola follows eq. (25). From ref. [26].

function of dimensionality. It is not clear what to expect for fmax and more generally for the shape ofthe scaling function f(z) for finite dimensionahities above six.
Universality for the cluster size distribution has also been tested in two [71] and three [26]dimensions. By adjusting the two lattice-dependent parameters q0 and q1 in eq. (21a) Leath and Reich[71]could confirm excellently the similarity of triangular and square lattices with respect to cluster
numbers, for s above 85. And in three dimensions [26],bcc and sc lattice also seem to have the same
shape of the scaling function f, as required by universality. Moreover, by usit~g(p p~)Ip~
instead ofp p~in the definition of the scaling variable z, the two lattices even could be described [26]by thesame parameter q1, a particularly simple form of universality (also perhaps not exact).
In conclusion, scaling for the cluster numbers n5(p), eq. (15), seems to be confirmed well forsufficiently large clusters close to p~in two and three dimensions. In two dimensions, different workby different authors using Monte Carlo and series techniques gives consistent results; deviations
seldomly amount to more than ten percent. A comparison of two lattices in twO and three dimensions
confirmed well the universality concept. Very little is published on cluster numbers for more than
three dimensions [45].
——3.2.4. Decay far away from p~,
In eqs. (23, 24) we noticed already that for p p~the cluster numbers decay exponentially for larges, with log n3 x
above p,~,and cc s below Pc. That conclusion was bas~don Monte Carlo datawith concentrations between Pc 0.05 and p~+ 0.05 rather close to the critical point. Away from thecritical point, the inequalities of section 2.4 give for this decay exponent: ~(p 0) = 1 and ~(p 1) =1 lid in d dimensions [90].How is the situation for intermediate concentrations, say at p = ~ And
what about three dimensions?
Monte Carlo methods for the cluster numbers do not work well far away from Pc for large s since
only close to Pc many large clusters appear. But since the typical cluster size sE cc Jp p~’~’is verylarge only close to p~’we now perhaps are no longer forced to go to very largó values of s to see theasymptotic behavior. Thus the exact cluster numbers of Sykes et al. [42,44] can be used for ananalysis. Far above Pc [46, 49, 50] the series data show, similarly to the Monte Carlo results of section
3.2.2, that a simple power law is quite good: log v, cc
fits for d = 2 and •3~dimensionsthe cluster
—

~1/2

—

—

~-*

-+

—

—

511,d

34

D. Stouffer, Scaling theory of percolation clusters

numbers from s = 1 to s = Smax -= 10 within a factor 10 or better, even if the cluster numbers vary overtwenty order-of-magnitude [49].This result may have implications for nucleation theory [38,118] ifthat experience can be generalized to thermal phase transitions. It means that in the cluster numbers
n5 a “surface term” or s~’~can be extrapolated down to very small s, even pairs and single sites,
without giving a wrong order-of-magnitude for n,. One only has to fit the “microscopic surface
tension”, i.e. the factor in front of
in log n,, on the true cluster numbers in the size range to beinvestigated [49, 118, 119]. Thus above p. the result ~= 1 — 1/d is established not only by the
Kunz—Souillard theorem [90],eq. (11), but also by numerical data for rather small clusters in two andthree dimensions [46,49, 50] and for larger clusters in two dimensions [26,62, 115].
Similarly below p~,series data for p sufficiently far below p~give log v., cc — s for not too small
clusters, i.e. ~= 1 [46,50, 70]. Since rigorous or reliable inequalities, section 2.4, cover the range fromp = 0 to p pA with their prediction ~= 1, and since PA is rather close to p, (e.g. pA = 0.41 in thetriangular lattice where PC = ~) these older data were obtained in a region where later inequalities
confirmed their result ~ 1 more reliably. But they show in addition that the asymptotic behavior can
51—I/dalready be observed for cluster containing only ten sites.

Summarizing the information reviewed here on the asymptotic decay of cluster numbers we foundfrom a variety of different methods that most probably~(0<P <Pc) = 1 and ~(Pc <p <1) = 1 — lid.
(26)
(Right at p~,the clusters decay no longer exponentially but with a power law: n~cc s~.)This result for
random percolation, eq. (26), agrees entirely with Binder’s suggestion for (appropriately defined)
clusters at thermal phase transitions [36,120]. It is therefore a sign of hope that Stoll and Domb [115]observed similar behavior above PC for correlated percolation, section 6.2.3 in two dimensions.
Perhaps some better understanding of clustering for thermal phase transitions [25,36, 155] at least in
two dimensions is waiting in the near future.
These results for the decay exponent ~have some implications which in the cluster literature are
discussed under the heading “essential singularities” [34, 36, 121, 122]. If we look at the generating
function or “free energy” F(h) = ~. n~e~”,we may ask if F(h) is analytic in h at h = Ot Or more
precisely: Does a Taylor expansion of F(h) work with F(h) = ~kFkhk/k!for small positive fields h?
(Here F,. is the kth derivative d’~F/dh”at h = 0.) Obviously all field derivatives F,, are finite for all~ J3c since they are given by the moments ~ s~n5and since the n. decay exponentially according toeq. (26). But what is the radius ~of convergence for this Taylor series? If it is finite then F(h) is
analytic; but if R~00~= 0 wecallh this2) at
=0.situation an essential singularity, similar to what happens with thefunctionF(h) =ratio
exp(—
1/h for power series gives ~The generalcriterion
as the limiting ratio of IF,,/k!IiIFk±l/(k+ l)!I.
Thus in our case we have ~= lim,,...4~.~~k+I,~x (k + 1). Evaluation of these moments forlarge k is easy [701since then only very large cluster sizes s are relevant, where we may use theasymptotic form log n~or sC. Thus we find ~ skn, f
5k exp(—const~s’) ds, where all p-dependent
factors are hidden
intheconstants.
Themaximumof log[s” isexp(.
. .)] isat sthe=tt’; the value of s” exp(...) at this maximum
roughlys~cclocatedk” since[k/(~
const)}~’
cckexponential term is then negligible. (Proof: log[s” exp(. . .)] = k~in(s) — const Sc = k . In(s) — k/~ln(s’~).The reader who mistrusts these tricks may evaluate the integrals by gamma functions.) Thus
the kth moment ~. s”n~varies for k-+oo as k’~’:The smaller the exponent ~ is the stronger is theincrease of the higher derivatives with the order of the derivative. Thus RCOflV equals the limit (forlarge k) of (k + l)k”/(k + ~= (k + l)’”(l + lIk)~” (k + l)’11’e~”cc k’(—1/k~’~’~
for
—

35

D. Stouffer, Scaling theory ofpercolation clusters~‘=

1—l/d); thus [70]=lim kl~I’c.
k-soc

Thus below the percolation threshold, where ~= 1, the radius of convergence is finite. But above
and at p~where ~< 1 for all finite dimensionalities, the Taylor series has zero radius of convergence.
This somewhat unusual behavior is called here an essential singularity and e*ists therefore above but
not below p~.(The reader may have guessed that result already by noticing that F(h) = ~ nse~ mayconverge for negative h only if log n, varies at least as —s and not if it varies only as _~i~2,forexample.)This extremely weak “essential” singularity has long been suggested [34,36, 121, 122] from clustermodels for thermal phase transitions (see ref. [123]for a renormalization group approach); the first
rigorous proof came from Kunz and Souillard for percolation [90].We see its physical significance[122]mainly in the implications it has for the exponent ~,i.e. for the behavior of very large clusters.
An essential singularity means that the cluster numbers cannot decay as a Simple exponential. The
behavior of these large and very rare clusters is relevant for nucleation theory [38].(For dilute
ferromagnets at finite temperatures, Schwartz [88]has shown the Griffiths singularity [124]to be an
essential singularity in this sense as a function of the magnetic field, if all moments of the percolation
cluster size distribution exist, as they do below p~,[125].)
The exponent ~in log n5 or s~discussed so far describes only the leading term in log n5 for larges. More generally we make the ansatz: In n, = const~s~ 0 . ln s that mear~s[49,42]
—

—

or ~

—

exp(—const~Sc).

(27)

Here, just as ~ earlier, 0 = 0(p) is a new exponent to be fitted on the data. For p -+0 we expect thisexponent to agree with the exponent 0 for “animals”, section 5, whose value is listed in table 2. Of
course, at p = p~we have 0 = r, ~ = 0. Series analysis [49]by a modified ratio method gave in two
dimensions for intermediate cluster sizes an effective exponent 0(p). It increaged from 0(p -+0) = 1 to0(p Pc) = r, first slowly, then rapidly. Above Pc it decreased again with increasing p, but theextrapolation [49]became erratic for p close to unity. The trend in the extrapolation suggest possibly
a simpler behavior of the true (asymptotic) exponent 0 for very large cluster sizes: Then 0(p) stays
fixed at its “animal” value of table 2 for all p between zero and p~and it jumps to 0 = r at p = Pc. (Insix and more dimensions we expect 0 = r = ~everywhere.) We know of no scaling laws relating this“prefactor exponent” 0 to the other critical exponents or the dimensionality; neither is r 0 a
constant nor is 0 = ~d in general. Apparently a new idea is needed here!
With this remark we conclude section 3 which we regard as the main section of this review. The
results in sections 4 and 5 are at the time of this writing less reliable and more controversial than thosein section 3.
—

4. Cluster structure
In the cluster numbers we saw in eq. (26) a clear asymmetry between n, abØve p~,and n, below p~.
Now we want to know if this asymmetry is reflected in the structure of clusters. We first discuss thecluster perimeter which can be derived directly from the cluster numbers, without any new assumptions~.Then we look at other properties like cluster radii which require additioi~alscaling assumptions
beyond eq. (15).

36

D. Stouffer, Scaling theory of percolation clusters

4.1. Cluster perimeter and internal structure

The average perimeter t. and the width & of the perimeter distribution are discussed here fromthe cluster numbers with the help of exact relations [71,90, 1261.
4.1.1. Exact relations

Let g,i be the number of different cluster configurations for s sites with perimeter t, where the“perimeter” (not necessarily a “surface” in the usual sense) is the number of empty neighbors ofof s-clusters withcluster sites, section 2.2. Then n5, = g5,p5(l —p)’ is the average number [39—41]
perimeter t, and n. = ~ n,~is the average total number of s-clusters. We define an average perimeter t.
and width i~. for the perimeter distribution function n,jn, by
1/2

= ~

t. n
5jn,;

(28a)

~ = [~~t — t,)2n~~/n5].

(One could also define averages by summing over s at fixed t; but little work has been done in thisdirection so far [48].)The evaluation of the exact n~,/n5shows a unique pronounced peak near t =[42].
The averages in eq. (28a) can be expressed by derivatives of the cluster numbers n5(p), just as instatistical thermodynamics one finds the average magnetization and its fluctuation from the first andsecond field derivative of the logarithm of the partition function. Let Z~= n5p’ = ~, g,~q’ withq = 1 p be the partition function for s-clusters. Then the reader can check easily that the following
two relations are exact in general, not only for large clusters near the critical point [90,71]2lnZ
t. =(8/a lnq)lnZ,;~=(8/t9lnq)
5.
(28b)
5 we can rewrite these results as
By differentiating n5/pt~= (q/p)s q(t91 ap) ln n
5
(28c)
—

—

2s+q2(o/ap)2lnn,—q(a/ap)lnn

~=

qp

5.

For large clusters and p

Pc

we have ln n~cc

(28d)
— ~

with a proportionality factor depending on p

and an exponent ~given in eq. (26). Thus for all p except at Pc we get for s ~:2s + O(s’).
(29)= (q/p)s + O(s’)~ = qpAccording to Kunz and Souillard [90]the exponent ~is rigorously below unity for all p above PC and-+all finite dimensionalities; hence
urn t,/s = (1 P)/p—(p ~ PC).

(30a)

Moreover, if ~= 1 l/d above p~,then the “correction term” cc ~ in eq. (29) varies as s’”~above
i.e. as a cluster surface area. Since the perimeter t, counts internal holes as well as externalboundaries, its proportionality to s for large s is legitimate; see also section 4.3.2. (For p = PC thevalidity of eq. (30a) will be shown in eq. (33).)
This relation (30a) has a lively history since its simple derivation [90]was long overlooked: Domb[101]predicted t, to be proportional to s for large s near p~,and Stauffer [98]suggested the factor of
proportionality to be (1 p~)/p~
at p~.Leath [73]gave a much simpler but still not exact derivation ofthat relation; moreover he gave the first Monte Carlo confirmation, albeit somewhat below Pc. Series
—

Pc,

—

37

D. Stouffer, Scaling theory of percolation cluslers
I

I

E

I

=10

I

I

I

I

F

.

-

I

0.8

++

-

0.6
0.40.2

-

.

-

P~
I

0

I

0.2

I

I

0.4

I

Ii

I

0.6

I

0.8

I•

1.0

P
Fig. 12. Variation with concentration of the limiting perimeter-to-size ratio for large percolation clusters. The solid curve gives the exact result(I — p)/p above p
5 [90]. The Monte Carlo results of ref. [62], +, and of ref. [75],dots, and the series result from ref. [52], x, refer to the squarelattice.

extrapolations by Stauffer [50]and Domb [102](but much more precise by Duarte [52])agreed witheq. (30a) at p~.For p slightly above Pc Stauffer [126]gave scaling arguments fo~eq. (30a); according tothat work eq. (30a) is not valid below p. A proof of eq. (30a) at Pc was given by Reich and Leath [89],and for all p above p~by Kunz and Souillard [90].An independent different aigument was presented
by Hankey [91]for the analogous result in the infinite network for all p above p~.Stoll and Domb [62]and Leath and Reich [71] confirmed eq. (30c) by Monte Carlo simulations~for p at and near p~(ref. [62]also for the infinite network), and Stauffer [75]added the Monte Carlo confirmation for p far
above p~.Below Pc deviations from eq. (30a) were observed by Monte Carlo [62,75] and series work([52] and Duarte, priv. comm.).
The corresponding result for the width of the perimeter distribution,2(p ~(30b).
lim&/s = (l/p)p
has a much simpler history: It was investigated theoretically and experim~ntallyby the Rutgers
University group [71—73,
89] whereas Stauffer’s series extrapolations [50,alsO 46] turned out to be
wrong [126].All this numerical work was restricted so far to two dimensions; fig. 12 summarizes thevariation of the limiting ratio of perimeter to size with s. See Note added in proof, Peters et al.
The partition function Z, = Y.~g,,q’ for an s-cluster corresponds, in statistical mechanics, to thepartition function Z = ~E g,~exp(—EIkB T). Here ~ is the number of differer~tquantum states for s
particles with total energy E. We see that the perimeter t for percolation is the analog of the energy E
for thermal phenomena; and if each broken bond between nearest neighbors cOntributes an energy 2J
to the E, then we may identify in this analogy the quantity q = 1 p with exp(—2J/kB) and theperimeter t with the number of broken bonds, such that the energy E correspOnds to the product 2:.!.
(Kasteleyn and Fortuin [21]give a more rigorous formulation of this analogy.) Apart from the trivial
factor 2.!, the ratio lim(t,/s) = (1 p)/p then corresponds to the thermal energy per particle, in thethermodynamical limit. Numerically this analogy is quite accurate [98]:For exa~nple,in percolation on
the triangular lattice we have p~= ~, lim t,/s = 1 at p~,i.e. to each critical cluster site belongs, in the
average, one perimeter site. Analogously, for ferromagnets (spin ~Ising, inter~ctionbetween nearest
neighbors only) in the triangular lattice at the Curie point the average thermal e~iergytells us that each
—

—

38

D. Stouffer, Scaling theory of percolation clusters

spin has, in the average, one of its six neighbors antiparallel [98].Thus the number of broken bonds
per spin approaches at the Curie point the same limit as the number of perimeter neighbors per site at
the percolation threshold, i.e. exactly one. Thus it is natural to expect [1011the perimeter t5 to be
proportional to the clusters size s, just as the thermal energy is proportional to the system size formost problems in nature (provided surface effects are negligible). In particular also forthe droplets usedinclassical nucleation theory [38]or in the Fisher droplet model [34]the thermal energy and similar
quantities (like entropy) contain a bulk term proportional to the number of molecules in the liquid“raindrop”. Broken bonds in the interior ofthe material are responsible for these proportionalities to thesystem size [50].
4.1.2. Scaling theory

Let us restrict ourselves now again to large clusters near PC where scaling assumptions,eq. (15),
Tf(z) andz=
can—p~)s~
be applied(p
we getto calculate the derivatives of n, required in eq. (28). With n, cc s
= (q/p)s + s°i/

2s
1(z);

with
=

—

q

d(lnf)/dz;—

s2°i/i2(z)

(31)

= qp
1//2 =—

q2d2(lnf)/dz2.

Thus the leading term cc s is very simple, but the higher order term contains a complicated scaling
function. We now call the difference t~= t, (q/p)s the “excess perimeter” [126]since above p~thisexcess perimeter ~ or
51-I/dseems to come from a surface area. On the other hand, the “bulkperimeter” (q/p)s may be interpreted as a volume effect.
8(I_l/d)xMore quantitatively, above p~with ln f cc
=
= (p
p~)P
51—l/d the excess
perimeter is t = s”çb~cc s~d(lnf)/dz = d(ln f)Idp or (~ ~) 1+$8(111d)5Eq. (37) will show us laterthat the volume V~of a very large s-cluster varies as (p —p~)~s
above p~[50]; then the excess
perimeter, expressed by the cluster volume, varies for very large s as (p p~’~”~ x ~
or
—

—

—

—

1—l/d

—

cc (p — p )(d_1)Pl(p > PC)~(32a)where the scaling laws (18b) have been used. Again we see quantitatively the analogy between ourperimeter and the energy for thermal critical phenomena: The excess perimeter has the samedependence on p
as the surface energy of a fluid or the domain wall energy of an Ising
ferromagnet has on T~ T. Thus we may identify— PC—

excess perimeter ~ surface energy (p > Pc).

(32b)

(Disscussion of the exponent: The surface area of a homogeneous liquid sphere surrounded by its
vapor varies as (volume)l_L~~~in d dimensions. Its surface free energy is the product of surface tension
and surface area. The surface tension, as reviewed e.g. by B. Widom in vol. 2 of ref. [11]vanishes as(T~ T)~1)c’ near
the critical
its temperature
gives theenergyper unitarea
2a(F/T)/DTor point;
T~9F78T.
Thus the derivative
surface energy
persurface
unit areavariesas (T~
since
E = which
T is the same exponent as derived in eq. (32a) for the excess perimeter.)T)~’~”~,
Right at p = p~,the derivative d(ln f)/dz is negative, as can be seen e.g. from the logarithmic plot in
fig. 11. Its value at z = 0 is estimated [261to be about —7.2 in the triangular lattice. Thus we have fromeq. (3la) [98]:———ts(PC) = s(l — p~)/p.~+ const~—39

D. Stouffer, Scaling theory ofpercolation clusters

where the constant is about 3.6 and o 0.39 in the triangular lattice. This relation is confirmed by bothMonte Carlo and series results [52,71]; e.g. the Monte Carlo work gave t5 s + 3.43 ~O.4o±o.oI and theseries work an exponent of 0.38 1 ±0.003, for the triangular lattice.
Below p. we have ~= 1 which means: No clear separation is possible between a surface term and abulk term in the cluster perimeter, since both vary as s, as shown by eq. (29). Thus a surface in theusual sense of droplet surfaces seems not to exist below p~,but only above Pc. Only above P~can weinterpret the excess perimeter t~as a “measure” of the surface area in the usual sense. Already at
P = PC this suggested proportionality of excess perimeter and surface area breaks down, as pointed
out by Domb and Stoll [97]and in contrast to our first speculations [98]:Eq.~(32b) indicates that theexcess perimeter really is a surface energy, not a surface area. As long 1”,
as thetensioniseq. surface(32a). But
right atindependentofclustersize,
both
surfacearea
andsurfaceenergyvary
assp~the powers of (pappearing in the surface tension above Pc are replaced by powers of s, andthen surface area, surface free energy, and surface energy no longer vary withthe same power of s. In
this sense the exponent o~is not a true surface exponent, although the term s°~fi1(z)does give above p~
the analog of the surface energy, eq. (32).
Also the external perimeter, which ignores all holes in the interior of a cluster, has been suggested[50]as a measure of the surface area for large clusters. However, Leath afld Reich [71]found forp ~ p~most of the perimeter sites to be external; fig. 13 shows one of their clusters with 4741 sites at
p = 0.48 in the triangular lattice. Many fjords connect the perimeter sites with the outside world [71].
At p = Pc ref. [71]found the external perimeter in the triangular lattice to indrease as
for cluster
containing several hundred sites. This exponent 0.9 is clearly different from t$e exponent o- = 0.4 forthe excess perimeter at Pc and also from the two-dimensional surface exponent I l/d = ~. Thus atleast at p~the external perimeter is not proportional to the excess perimeter, contrary to a hope in ref.
[126].Also far above p~,[75] most of the perimeter sites are external. (Perhaps in the square lattice[75]at p = 0.75 for s above l0~the internal perimeter roughly equals the bu’k perimeter (q/p)s, andthus the external perimeter approaches the excess perimeter (surface energy)’ for very large clusters.
But since clusters with more than 1000 sites are very difficult to produce this, result would still meanthat the external perimeter is not a practical tool to study cluster surfac~s.)In three and moredimensions we have p~,<~ in general; thus near PC an infinite network of holes percolates through thelarge clusters of occupied sites. Then the external perimeter completely loses its meaning as asurface measure.
Therefore at present the excess perimeter t~= t5 (q/p)s = solI,i(z) seems to be above p~the mostpromising measure of a surface area for large clusters, eq. (32b). Monte CarlO data have shown thatt~/s’~
is indeed a function of z = (p p~)s~
alone; data for different p p~fol1~wthe same curve [71],as in our scaling plots for the cluster numbers. The cluster size s had to be larger than 85 to agree withscaling [71].Moreover, since eq. (28c) is exact, Monte Carlo data for the excess perimeter t~can beused to find cluster numbers n0. In this way, and not directly from the n, in ~1g.10, Leath and Reichfound their data [71]to support ~= 1 against ~= 2c for large clusters above j~.
The width & of the perimeter distribution function is right at p, given by ~q.(3lb):
2sconst= (1p~)p~
1 ~2cs + const2 s°~(34)
The correction exponent 2o’ here is about 0.8 in two dimensions and even clØser to unity for higher
dimensionalities. Thus even for rather large clusters the leading term is not m~ichlarger than the firstcorrection term. For example, if o = 0.48 in three dimensions, then only foit s above 10~sites per
cluster would
be less than ten percent of s. Even if every human being ~n earth would have its
— PC)

NO.9

—

—

—

—

—

~2o

—

40

D. Stouffer, Scaling theory of percolation clusters

cc

LI-.

~
I

j

I

‘

.~-

r

~

I~
.~

~

I

Is

~
~ ~4 ~I~.~-;~i
‘~~
.N~
~-

~

L

Ii
-- ..4•~~~..: I—~

,°~

‘-..

s..’

—.

~~‘‘

I

4

h~o

+~

~

~

~

~

‘

:
~

~..

4.-c~c-,-.

..

~

~

~

.--.~.-.-~-

~

~

~

~

~

I

‘~~<:

~

-~‘.

Fig. 13. Example of a large cluster slightly below p5. The sites are connected by lines and are surrounded by the perimeter sites °. From ref. [71].

41

D. Stauffer, Scaling theory ofpercolation clusters

own computer to store l0~sites [24]the combined memory would be too small by many orders ofmagnitude to handle such clusters. Therefore the correction terms in eq. (34) cannot be neglected.
Series expansion analysis which neglected the correction term gave wrong exponents [46,50]. Even in
two dimensions, where the situation is better, Monte Carlo data [71]show a strong deviation from theasymptotic law, eq. (30b) even at s = 2000. However, with the two constants in eq. (34) determinedfrom the cluster numbers in the triangular lattice [26]via eq. (28d), the Monte ‘Carlo data of Leath andReich for the width [71]fit nicely with eq. (34) for large s.
Thus the width ~, of the perimeter distribution function varies as s112 the mean value t~as s, andthe excess perimeter t~as s°’ multiplied with a scaling function ‘//~. Because of these different
exponents we have here an example where scaling assumptions do not hold. To describe the functionn,(p) of two variables s and p p~we assumed n~cc sTf[(p p)~U] The analogous assumption for
the three variables s, p Pc and t entering the perimeter distribution would be———flst(P) or s~”f1[(p pc)Soc, ts°]
(35)
with a new phenomenological exponent v. (u = ypsilon; not to be confused with v = nu for thecorrelation length, eq. (6e).) (The prefactor s~” is chosen such that summation over t gives thedesired n,(p), eq. (15).) Such an assumption has in fact been made for the energy distribution of Isingmodel clusters [36]and the bond distribution of percolation clusters [114].‘But it is wrong for the
perimeter of percolation clusters. To accomodate ~, cc s, we need u = 1; to allow ~, cc
we needv = ~: Contradiction. If instead we postulate eq. (35) to be valid for the excess perimeter t~=(q/p)s cc s°,we would need v = o~to accomodate this excess perimeter, but still v = ~ for thewidth: The contradiction is not removed. Only close to six dimensions, or in the classical regime
above six dimensions, the contradiction
vanishes fortheweexcesssinceseethencr =~apart from
2 below 6 dimensions.
Thus
see, perimeter,
just as we willwithuniversality
inaappendix
correction
of
order
(6—
d)
2, that scaling is valid in most cases but not in all. The numerical tests of section 3.2 reallywere necessary and did not just prove a triviality.
—

~h/2

—

Internal structure
Returning to bulk properties proportional to the cluster size s, we find besides the limit of t,/s also
other quantities characterizing the internal structure of large clusters. Trivially the concentration p is
one of them. The “energy” e, can be defined for s-clusters as the average number of nearest neighborconnections between an occupied cluster site and an empty perimeter site [48,60, 127]. The bondnumber b, in s-clusters is the average number of nearest neighbor connections between cluster sites[114].The cyclomatic number c, is the average number of cycles formed by, the bonds in s-clusters(i.e. by its nearest-neighbor connections); e.g. in a Bethe lattice the cyclomatic number of trees iszero, for a polygon it is unity, and it is largest for fully compact configuratiofls without internal holes[97,115]. The genus g, is the average number of topologically separated holes in the interior ofs-clusters [128].For every configuration, and thus also for the averages, on~has [97]c, = b, s + 1
and e, = (coordination number)~s 2b,. The compact 4-cluster 0 for example has a perimeter of 8, an
energy of 8, contains 4 bonds, has cyclomatic number one, and genus zero. FOr large clusters all thesequantities vary as s; for example lim,..,,.(g,/s) = 1/28 in a bcc lattice [128]at Pc. Refs. [115,97, 1291
give the monotonic variation of lim(c,/s) with p. Besides these Monte Ca~loresults also analytic
results are available, as reviewed by Domb [129]. So far no scaling analysis i~iterms of exponents orfunctions of z were given for the leading terms or the first corrections; Su~et al. [60]remark thate, = const
1 . s + const2 according to their Monte Carlo data. In that respect thó perimeter is at present
4.1.3.

——

42

D. Stauffer. Scaling theory ofpercolation clusters

better understood and investigated [71]than cyclomatic number, energy, bond number or genus in
their scaling properties. (Refs. [97, 115, 129] also compare these bulk properties of percolation withthose of ferro- and antiferromagnets at finite temperatures.)
In conclusion, the internal structure of large percolation clusters in terms of perimeter per site andanalogous quantities seems well studied by now. For the perimeter also the leading correction term
was studied in detail and found to agree with scaling theory. A cleary asymmetry about p, wasobserved in this excess perimeter t,~ (qlp)s and is related to the asymmetry in the cluster numbers.
—

4.2. Radius and density profile

4.2.1. The basic assumption

To describe parameters affecting the external shape of clusters, not their internal structure, newquantities like density or radius seem necessary. They are defined as statistical averages over a cluster(and, of course, also as averages over many clusters of the same size) and thus do not give us
information on microscopic details like cyclomatic numbers etc. [115].To describe their scaling
behavior a new “hyper”-scaling assumption seems necessary. This assumption then will give us thescaling law dv = 2— a postulated in eq. (18b) but not yet derived.
Eq. (14) gives us a hint how a scaling assumption similar to eq. (15) for n~may be constructed forthe cluster radius R~:The ratio R5/~might just have the same scaling property as n~(p)/n5~or
i.e. it may depend on the ratio s/se only. Since s/se or Z”~with z = (pns(p)/ns(pc),

—

p~)s~
we thuspostulate
R, =

(36a)

. R(z)

where 1~is a suitable scaling function to be fitted on experiment. We may rewrite this assumption as(36b)

R. = 5ocv1~(Z)since ~ cc p ~cc sr. In this form our assumption [130]has the same structure as eq. (15).
We now assume in addition that the internal structure of a very large but finite cluster above p~isthe same as that of the infinite network. “Very large” means here: sIs~ 1 or z 1, that means radiimuch larger than the coherence length. “Internal structure” refers to average properties like density,
visible over distances much smaller than the cluster radius. Thus we postulate:
—

~‘

~‘

By looking at average cluster properties over distances much smaller than the cluster radius wecannot distinguish between very large but finite clusters and the infinite network, if p is above p~.
(36c)
An example for the validity of assumption (36c) is the ratio t5/s for very large clusters, which is(1 p)/p according to eq. (30a). In the infinite network, the corresponding ratio has the same value[62,91]. Thus by looking only at a finite fraction of a sample we cannot tell if the large set ofconnected sites which we might observe there extends to infinity or belongs to a finite cluster.
Similarly if a man and a woman get along well for one evening they still might not make it together
through a long marriage. In that sense eq. (36c) is compatible with numerous experiments. But it isobviously constricted to concentrations p above Pc since below Pc there is no infinite network present
with which very large but finite clusters could be compared.
—

D. Stouffer, Scaling theory ofpercolation clusters

43

4.2.2. Exponents for the radiusWith this assumption we now get information on the limiting behavior of R3 above p~,i.e. on thescaling function R(z) or R1(z) for z oo. In every unit volume of their interior the very large clusters
thus should have the same fraction pP.. of sites connected to the cluster as ha~the infinite network. In
short, the average “density” inside very large clusters is the same as the density of the infinite
network. (The density is defined as the probability that a given lattice site belOngs to a given cluster or
network.) If surface effects are negligible for these very large clusters, we can calculate the cluster
volume V5 from the “mass” s and the density pP.. through the well known relation: mass = volume
times density:
s=V5pP..;
p>p~.
—~This argument was called the “Swiss cheese” picture in ref. [50],since in spite of its many internalholes a very large chunk of Swiss cheese still has a mass. s proportional
0,we thus getto its volume V~.Close to p~,
where the percolation probability vanishes as P.. cc (p p~)—Vs (p —p~)~ (s ~ s~, p >pj.

5

(37a)

Note that neither the cluster radius R5 nor the cluster volume V5 were defined here quantitatively; forexample, one may use for R. the radius of gyration: R~= ~, r~/swhere the sum runs over all clustersites, and r1 is the distance of a site from the center-of-mass of the cluster. We assume that this and allother “reasonable” definitions give the same critical exponents. We may de~1nea cluster volume inthree dimensions through V11 = (4w/3)R~,and in d dimensions we have V5 cc R~.Thus eq. (37a) leads
toR5 or (p _p)O/d.
(37b)
for very large clusters above p~.
If both eqs. (36) and (37b) are assumed to be correct, then the function R(z) in eq. (36a) must vary
as zlIb0’~I~ ~1Id for z—*x, in order to make the radius proportional to slld,in accOrd with eq. (37b). Now
we get from eq. (36a), for p slightly above p~:(p pc)u~~~f511d cc (p _p~)_v+lbad. 51id~
If we compare this exponent of p p, with the exponent appearing in eq. (37b) we conclude/3/d v herd, or dv = /3 + 1/er = /3(8+ 1). This is the desired scaling law of eq,. (18b) whose derivation
was missing so far. Alternative derivations were given in refs. [18]and [109].
This scaling law dv = /3(8 + 1) = 2— a is also known from other phase ttansitions [10,111. It isregarded as less reliable than the other scaling laws not involving d, but presui~iably[111]it is exact ora good approximation for not too high dimensionalities d. For d ~, on the Other hand, one expects
classical exponents a = 1, /3 = 1, y = 1, 8 = 2, v = ~, as in mean field theorie~[114]or Bethe lattices[27—30].
For all d above six, the above scaling law dv = 2— a breaks down if these classical exponents
are used. Therefore Toulouse [84] suggested, as confirmed by later resi~lts[45, 68, 117], that“hyperscaling”, dv = 2— a, is valid only below six dimensions whereas classical exponents are valid
above six dimensions. Only at the marginal dimensionality d =6, where the t~oregimes coalesce, areboth classical exponents and hyperscaling valid; but then logarithmic correcti4n factors are important[17].This special status of six dimensions as the transition from the cIassi~alto the hyperscaling
regime is the reason why the epsilon expansion of renormalization group is a perturbation expansion
about 6— d. For thermal phase transitions the classical exponents are usual~y/3 = ~ and a = 0 (and
cc ~.

—

—

-+

—

D. Stouffer, Scaling theory of percolation clusterso = 3); thus the same arguments as above give four dimensions as the marginal dimensionality above
which classical exponents are valid. From now on we restrict ourselves to dimensionalities below six
and assume that hyperscaling is valid, i.e. 2— a = dv.
So far we dealt with p above p. only; but the scaling assumption (35) should be valid for both sidesof the phase transition. Right at p~the average cluster radius R5 is finite and thus cannot be
proportional to any power of p Pc. Thus in eq. (36b) we have R1(z 0) const, or R5(p~)cc
Since erv = dv/d/’38 = (1 + 1/6)/d = 2/(d + y/v) according to hyperscaling, eq. (18), we see that [50]—R5(p~)cc 5(l+118)/d;

—~

—~

(38)

V,(p~)cc

A different derivation which does not use hyperscaling explicitely was given by Harrison et al. [109].
(De Gennes, private communication, pointed out that with the approximation y 2v the above
relation reads R5 cc
similar to Flory’s approximation R5 cc s
for the radius of self-avoiding
walks [131].)
Harrison etfrom
gave the Montefirst andCarloso farbest confirmationof eq. (38).
Theyconcluded
3al. [109]also
three-dimensional
simulations,this exponent
2.66 isroughlycomspatibleor R~~°’
with the predicted value dI(1 + 1/8) = 2.50 ±0.07, from eq. (38) and [55] 8 = 5.0 ±0.8. Clearly
the exponent is not just equal to the dimensionality d. (For d = 2 the numerical evidence is lessimpressive since 6 is much larger there [75].)For thermal phase transitions eq. (38) has been proposedmuch earlier [74, 132] but no direct confirmation seems to exist; again percolation turned out to be thebest-understood cluster problem.
Below PC, if R. is assumed to vary for very large clusters as S” with a new exponent p, the scaling
assumption (36a) gives [130]3/(d+2)~2/(12~I.2)

R. cc p ~

(39a)

~

This result describes also very large clusters above p~and at p~,if the above values for the exponentp, eqs. (37b, 38) are used. It seems likely that the exponent p below p~is constant for all p between
zero and p~,just as the exponent ~was. If we call this constant v
0 we can summarize our results forthe radius exponent as
R5ccs” (s_s.xatfixedp)(39b)p(O <p <pc) = v0
p(p~)= (1 + 1/8)/d;P(PC <p <1) = lid.
(39c)
Figure 14 summarizes Monte Carlo results for two-dimensional cluster radii. Above and at Pc thedata are consistent with the predictions p = ~and p = 0.53 = ~(l+ 1/6), shown there as horizontal lines.
Below PC we find the new exponent v~to be about 2/3, significantly lower than the “self-avoidingwalk” prediction [130]of 3/4. It seems that a better theory [133]is needed since percolation clusters
below p~do not behave similar to self-avoiding walks.
The data of fig. 14 do not give exponent values which are constant in certain intervals and jump
discontinuously at Pc; instead the numerically determined exponents vary continuously with p. But we
see no real contradiction to the constant exponents predicted in eq. (39c). For these Monte Carlo data
are taken for clusters containing ten to thousand sites, whereas eq. (39) refers to very large clusters
only, s ~. Only for clusters much larger in linear extension than the coherence length or~
can the simple power law (39a) be expected to hold. Thus particularly the data close to p~should not
be relied upon since there the scaling variable I~ p~socis not much larger than unity [130].Therefore
only the data for p close to zero or unity. and for p = p are reasonably reliable, whereas the
exponents at other concentrations are effective exponents and not yet close to the true asymptotic
~

-*

—

—

D. Stouffer, Scaling theory of percolation clusters
p
0.7

I

I

I

~.6

~8

45

~

0.6
.
1(1+1)

0.5

~2

~4

p

Fig. 14. Variation with p of the effective exponent p for the cluster radius ?. Monte Carlo data from ref. [75],., ref. [73],x, ref. [109]and priv.
comm., +. The horizontal lines symbolize the expected results, which are discontinuous at p
5.

exponents predicted in eq. (39). Similar variation of effective exponents are known from thermal
phase transition; the analogy is particularly clear with self-avoiding walks with interaction [135].
Figure 15 tries to transform the above argument into a simple picture, analogous to dynamical
critical phenomena near Curie points. The asymptotic exponents of eq. (39c~are valid only in thesmall sectors I, II and III of that figure. In between these sectors the cluster ra~liusis described by thefull scaling functions of eq. (35), which only in the asymptotic limits of the shadowed sectors become
simple power laws. Similar effects are expected for other exponents like ~ and 0. Thus if we keeps = 102 constant and move continuously from p = 0 to p = 1, then we start in region III where p =1. With increasing p we leave region III and move along the dotted line of fig. 15. Here the cluster
radii and cluster numbers are described by complicated scaling functions like eqs. (35, 15) or evenmore sophisticated expressions. Very close to p~we reach sector I where p = (1 + 1/6)Id and n11 cc
Further increase of p along the dotted line brings us out of the sector I into the less understoodintermediate region, until we reach safer ground again for (p — p~)s°~ 1 in region II, where p = l/d,~= 1
lid is valid. Nowhere along the dotted line is, for fixed finite cluster size s, a discontinuity inthe cluster numbers or the cluster radii, since only the asymptotic exponents depend discontinuously~‘=—

Fig. 15. Variàus sectors of the s—p-plane in percolation scalingtheory. The shadowedregions correspond to a ‘~s~(s~ctor1), s ~ se above p5 (sectorII),and s a~.5~below p5(sector III). Only in these narrow sectors are the asymptotic exponents ~ 0 and p directly obserl~ablefor largeclusters. (The threesectors were drawn with an enlarged angle to make them better visible.)46

D. Stouffer, Scaling theory of percolation clusters

on p. A clear example of the similarity of small clusters above and below Pc is shown in ref. [115,fig. 101; it contradicts in no way our fig. 15 and eq. (39c) here which try to describe the limit s
only.
In conclusion it seems that the cluster radii behave roughly as they should. But higher accuracy anda better theory for v0 2/3 (for d = 2) are needed.
—~ ~4.2.3. Density profiles
In order to understand better the difference in the cluster radii and cluster numbers above andbelow p~it is useful to look at the density profile of clusters [75, 130, 134]. Let the density profile
D5(r) of s-clusters be the probability that a lattice site at distance r from the cluster center-of-mass
belongs to that s-cluster. (Contrary to ref. [130]we do not require the center-of-mass to belong to thatcluster; that definition would be needed for correlation functions which are not discussed here.) Thisdensity thus is a “coarse-grained” average [115],similar to other statistical concepts in hydrodynamics(density, velocity, pressure). It does not give us microscopic details on the structure of clusters;
instead the density profile describes the overall shape of a cluster.
As mentioned in the discussion after assumption (36) we take the density D5(r 4 R5) inside verylarge clusters above p~,to be the same as the density pP.. of the infinite network, i.e.
lim D5(r) = pP..

(40a)

for fixed p above p~,and fixed r. If we are not in this simple asymptotic limit, the density profile D5(r)depends on three variables r, s and p. In the scaling regime of large clusters near Pc we may again
postulate a scaling-homogeneous form [130]reducing the number of independent variables from three
to two:D5(r) = P.. 15(,l~,5/5~)(40b)or equivalently,
135(r) = P.. D2(rs°~,(p p~)s°).

(40c)

—

The interested reader may invent many other forms [130]of this assumption. Trivially we havefD5(r) dr = s since the integral over the density gives the total mass s. (Discussions of correlation
functions suggest for r ~ R5 and s ~ s~ the asymptotic decay laws log(D5) cc r above p. andlog(D5) cc r”°~’°~
below p,. [1301.)Thus if eq. (40c) is rewritten as D5(r) = s_xJii(r/s~~~~,(p —p~)s°)then we must have x = dp 1 in the three shadowed regions of fig. 15 [76].
This scaling assumption formulates more precisely the assumption made after eq. (37a) that all“reasonable” definitions of a cluster radius give the same exponent. Eq. (40) means that for a given s
and a given p there is only one characteristic length R5 cc s°~for the cluster, and not several of them.
Any reasonable definition, like R~= f D5(r) r” this, then will give the same length, apart fromconstant factors. Leath [73]used this definition with k = 2 in the first study of cluster radii, and was
followed in ref. [75].Harrison et al. [109]used instead of this “radius of gyration” the spanning lengthof the cluster, which corresponds, in some sense, to k = ~. As we saw in fig. 14 these different
definitions give roughly the same exponent in Monte Carlo simulations.
A complete test of the scaling assumption (40) is not yet known to us. Herrmann [76]looked at thedensity profiles [75]right at the percolation threshold p = Pc (and also for the “animals” p = 0) where
eq. (40c) reduces to the simpler form:———

D5(r) = s”~153(,1R5).

(41)

47

D. Stouffer, Scaling theory ofpercolation clusters

~

21J

I

~ 1.5

~

I

‘

0~~p10O

T

10

I

I

______________

I~ ~~ ~l~

D11(r)~~i- pP~

20 riRS25

I

r

Fig. 16. Plot [76]of the scaled density profile D,(r)/D,(R11) as afunction of the scaled distance rIR, at the percolation threshold(solid line) and for “animals” (p = 0, dashed line). Error bars

Fig. 17. Monte Carlo data [75]for the density profile far above andfar below p5 (averages over more than l0~configurations at fixed s

indicate statistical fluctuations as well as variations for different
sizes s.

but not below p5.

in the square lattice). These data suggest a droplet structure above

He found this similarity law for the critical percolation clusters to be consistent with two-dimensional
Monte Carlo data; in particular the prefactor was found [76]to vary as ~ -o.O5~±o.O25= ~ x confirming
well that x = dp 1 = 1/6 0.054 according to eq. (38). Figure 16 summarizes ~hescaling form of thedensity profiles. For p away from p~,fig. 17 shows the density profiles of two la~geclusters; again data
far away from ~ give the asymptotic behavior better than data closer to Pc is s is fixed at someintermediate size. The cluster density profile above p,~has a plateau in the cluster interior, with theheight near pP.., and decays rapidly to zero outside the cluster. The interior ~mnd
exterior region are
separated by a rather narrow interface. (Presumably the thickness of this inte~faceis of the order ofthe coherence length ~ [126];indeed already at p = 0.75 the transition is much smoother than at
p = 0.95.) Below p,, the cluster shape is entirely different for very large chusters: The plateau is
replaced by a bell-shaped curve [75,76] where the transition layer separating interior from exterior
parts now extends over the whole cluster. Thus a “surface” layer in the usua’ sense exists above p~
but not below Pc, if s —* ~. We think that this difference is the reason for the asymmetries between
above and below Pc discussed earlier in the cluster radii (p = versus p = ~),the excess perimeter t~(ccversus ccs), and the cluster numbers (~= versus c = 1).
In this way a coherent scaling picture has been developed and partially tested. The cluster numbers
n3, eq. (15); the excess perimeter t~,eq (31); the radius R3, eq. (36b); and th~density profile D5(r),eq. (40) are all expressed by “generalized homogeneous functions” [100],i~e.the number of independent variables is reduced by one:—

~1/2

cc s~f(z),~ cc s°.fr1(z), R. cc s011~1~i(z),D5(r)cc s~aOi(,1s01l,z).

(42)

And in addition to this mathematical similarity, all four scaling functions in eq~(42) seem to exhibit apronounced asymmetry in their asymptotic behavior for z -~±~, an asymmetry not restricted to p
close to p~but missing in classical theories (Bethe lattice).
4.3. Droplets, ramification, and fractal dimensionality

Headlines in newspapers try to inform the reader in a few words about the main content of thenews; of course sometimes they are misleading and one needs to read the full text. Similarly concepts
like ramification [101]etc. are simple catchwords trying to encompass as he~dlineswhat we have

48

D. Stouffer, Scaling theory of percolation clusters

described in greater detail in the preceding sections. And again these simple concepts may lead to a
picture oversimplifying what we said before. (The contents of this subsection are not needed later.)
4.3.1. The raindrop or Swiss cheese model
Since the Fisher droplet model [34]of thermal critical phenomena and the raindrops of classicalnucleation theory [38]long preceded the current interest in percolation clusters, it seems appropriate
to start with the question: Are percolation clusters similar to raindrops, as assumed in the Fisher
model above pc?
How does a raindrop look before it is falling on your head? It consists of s “liquid” watermolecules inside, and is surrounded by water vapor outside. The liquid interior is separated from theoutside vapor by a narrow transition region or surface layer, with the ratio of layer thickness todroplet radius going to zero as the radius goes to infinity. In the interior, the density, entropy permolecule and thermal energy per molecule are the same as for bulk water if the droplet is large
enough. More precisely, the density etc. of the raindrop interior is the same as that of bulk water at
the same temperature and not the same as that of a (hypothetical )T = 0 groundstate. In particular, at
any given moment the liquid water has density fluctuations which produce holes inside the raindrop
just like holes in Swiss cheese. Only if we average over these holes we get a homogeneous density, forboth raindrops and Swiss cheese.
Figure 17 and other results of the preceding section suggest that percolation clusters are likeraindrops above Pc but not below p~.Just like a Swiss cheese, they have on both sides of the phasetransition an average density lower than the maximum possible density; above p,~their interior density
approaches pP.. whereas it seems to go to zero for large clusters below p~and at p, [75,76]. Only
above p,, a relatively narrow interface region could be found. Of course, since a water droplet of
0.1 mm radius contains about 1017 molecules and the clusters in fig. 17 only 200 sites, the ratio ofinterface thickness to cluster radius is still quite large for our percolation clusters in comparison withraindrops. The reader should also keep in mind that the roughly spherical density profile of raindrops
and percolation clusters arises only by averaging over many different configurations. For a single
configuration, the holes and surface roughness of the Swiss cheese structure give a rather different
picture which even above p~is not dissimilar to fig. 13.
Instead of this definition of “droplet-like” by the density profile one may also call a cluster
droplet-like if for s —*~ at fixed p its surface area (measured by R~’’or t~)varies as 5111d if its
radius varies as s”” or its volume as s, or finally if the logarithm of the cluster numbers varies as—
lid With all these definitions we still find the same result: Very large percolation clusters (R5 ~‘ ~)are droplet-like above p. but not at and below p~,as the reader can check himself by going through thepreceding parts of this review. If one likes to use a word (different from “ramified”) to describe the~Fig. 18. Hungryhydra (clusters below p~)eats Swiss cheese (clusters above pj. The coherence length ~ is shownschematically. This review is based on
the assumption apparent from the figure.

D. Stouffer, Scaling theory of percolation clusters

49

opposite of droplet-like one may try “hydra-like” [50]:The tentacles of this monster will engulf thescientist who tries to apply the simple droplet picture even below Pc.
In this sense, percolation clusters larger than the coherence length seem to be droplet-like, similar
to a large sphere of Swiss cheese, for p above p~but not at and below Pc, a conclusion summarized in
fig. 18.
4.3.2. Ramification

Domb’s suggestion [32, 101] that percolation clusters are “ramified” started the development ofcluster scaling as presented in this review. Clusters were called ramified if theit surface area increases
with size s —~ simply as 5; they were called compact if they increased witl a smaller power of s.
What is the “surface area” for percolation clusters? If one takes the total perimeter as a measure of
the surface area [101]then all clusters are ramified; internal “surfaces” are in~ludedin that definition.
If the external perimeter only is identified with the surface area then the questiOn oframification is not
yet entirely solved for two dimensions. With the excess perimeter t~as a me~sureof the surface theclusters are ramified below but not above p~or at p~.Finally, with R~’as the definition of surface
area the clusters are never ramified in two dimensions. One may also use the density profile andthickness of the surface layer as criteria for ramifiedness; then one is likely to end up with ramified
being identical to hydra-like and compact being the same as droplet-like. As di$cussed above, clusters
are in this sense ramified below and at Pc but not above p~.(If R~’is taken as a surface area, it may
increase even faster than s with increasing 5; e.g. in six dimensions we expect R~ cc ~S(I+I/8)/6 = p5/4
at p~.Such exponents larger than unity for the surface area are nothing unusual: The smallest sphereenclosing a random walk with s steps on a lattice has a surface area cc~~~12in d dimensions.)
Later [97,115, 129] Domb defined ramification more as a quantitative than a qualitative concept by
regarding the cyclomatic number per site, lim~...,,(c,/s),as a measure of compactness for large clusters.
More precisely, c
5 was normalized [97]by the maximum cyclomatic number achieved in a fullycompact cluster, without holes or surface roughness. The degree of ramification is then equal to unity
minus the degree of compactness in this normalization. This ramifiedness decreases continuously withincreasing p, as shown by Monte Carlo data [115],without any dramatic change at p~.Ramifiedness inthis sense therefore is drastically different from other criteria which may change from “yes” to “no”
at PC, as listed above. Clusters in the Ising model of ferromagnets at the Curie point were shown [129]to be less ramified than percolation clusters at Pc. If one uses the perimeter pe~site, lim~....(t~/s),as ameasure of ramification one arrives at the same conclusion, fig. 12: Clusters below p. are more
ramified than above p~,and at Pc they are more ramified than ferromagnetic Ising clusters at T~[62,75, 115]. (If the energy per site [32],lim(e,/s), is taken as a measure of ramification for Ising
ferromagnets, and the perimeter per site as measure of ramification for percola~ion,then Ising clusters
are about as ramified as percolation clusters at their respective critical pdints [98].)Of course,
ramification in this quantitative sense [9,7,115, 129] is not a completely new concept but merely one ofmany quantities measuring internal structure and disorder, similar to the therm~lenergy per moleculeor the entropy per cubic centimeter.
4.3.3. Fractal dimensions

Mandelbrot[136] introduced the word “fractals” to describe objects with fractal dimensions d’smaller than the Euclidean dimensionality d of the underlying lattice or space. This concept, which isan application of the Hausdorif—Besicovich dimension, was applied by Leath ~opercolation clusters[73].Earlier, Reatto and Rastelli [134]in a discussion of the density profile of Ising clusters, used the

50

D. Stouffer, Scaling theory ofpercolation clusters

effective dimensionality d’ = i/o-i’ but did not call it by any special name. For general background on
fractals we refer to Mandelbrot’s books [136];percolation clusters are discussed more explicitely by
Stanley [106](see also recently Mandeibrot [137]).Roughly speaking, if the mass (or size s) of asystem varies as (radius)’~’then d’ is the fractal or effective dimensionality and can be different fromthe Euclidean dimensionality d; in particular d’ need not be an integer. We now review many different
definitions, some of which giving the same result, and denote them consecutively by d~,d~Mandelbrot [136,p. 196] used a standard method for estimating the fractal dimension: He divided alarge square lattice into
equal squares, counted the number N of those that are intersected by thelargest cluster, and determined d~through N cc ~ (However, the value 1.78 that he found on p. 197
is presumably biased: B.B. Mandeibrot, private communication, November 1978.) A second standard
method for estimating d’ consists in taking circles or spheres of increasing size and measuring theircontents or “mass”. Forrest and Witten [138]estimated in this way the fractal dimension d~of
two-dimensional electron-microscope pictures of large smoke particles consisting of many small ironspheres of uniform size. They counted the number N, of small spheres in regions of linear extent 1,
and found N, cc 11.6, giving a fractal dimension d~ 1.6 which is clearly smaller than the Euclidean
dimensionality d = 2. The latter would be found if the smoke particles would be spatially homogeneous and “compact”. These dimensions d~[136]and perhaps also d~[138]are fractal dimensions,determined in a way close to the mathematical definition [136]of that concept. The other definitions
listed below are less directly related to these mathematical procedures and are thus called “effective”
dimensions in this review. Perhaps when this paper is published the relations between fractal andeffective dimensions will already be clearer than at the time of this writing.
For percolation clusters at PC the cluster density profiles D3(r) are all similar to each other according
to eq. (41) and depend mainly on the ratio r/s°~,or equivalently on s/r”°”.This similarity law suggests[134]to call i/o-v an effective dimensionality d~,since we may regard s as the “mass” of a cluster.
Hyper-scaling, eq. (18), gives i/o-v = d ~/v = d/(l + 1/6). Thus
62

—

—

d~(p~)= d/(1 + 1/6)

(42a)

is the effective dimensionality at p = Pc as determined from the density profile. Also Harrison et al.
[109],Kunz and Payandeh [79]and Mandelbrot [137]use this combination of critical exponent 6 andlattice dimensionality d as the effective dimensionality 1or
fractalMore= generally
we maywhere
thisdimension.
effective d~(p)i/p is definednottranslateourdefinition
R.
cc
s”,
eq.
(39b),
into
s
cc
(R~)”
only at or near the percolation threshold and differs in different regions for p:d~(p> j,,~)= d;d~(p= p,~)= d/(i + 1/6);d~(p<pa) = 1/p

0

(42b)

according to eq. (39c); the last expression is about 3/2 in two dimensions [75].
In a different sense, Stanley [106]looks for a d’ relating the average cluster size S with the averagecluster radius R through S cc R d’ for p -~p~.(The above definitions instead were based on s
atfixed p.) How should one define these averages over all cluster sizes? (See also appendix 1.) If we use
our “typical” cluster size sE and typical cluster radius R,~for S and R we get an effective
dimensionality identical to d~(,p~),eq. (42a). If instead we define these averages through S =~ sk+I,1j~ ?n~,R = ~ R3s~’n3/~~
5k~ we first have to distinguish whether we mean the full sums
or only their nonanalytic parts. The singular parts give S cc s~and R cc R3~= ~ and the resulting
effective dimension d~with s cc R”~is the same as d~(p~).
If instead we do not subtract the analytic
background from the sums, then we get again the same result for k = 2, 3, 4,. . . (provided d is smaller-*~

D. Stauffer, Scaling theory of percolation clusters

51

than 8 + 1, which is the case for d = 2, 3 and 4). But for k = 0 and k = 1 different exponents result(details of how to evaluate such sums are explained in appendix 1): Close to the percolation threshold
we have then S = const, R = const for k = 0, and S cc Ii’ PCfl, R cc [P PCI~~
for k = I. Thus withk = 0 we cannot define an effective dimensionality, and with k = 1 we get d~= y/(v — f3) which equals 2
for two dimensions and about 4 in three. Stanley [106]chooses k = 1 for the average cluster size s,
which thus diverges with the exponent y, but takes the typical cluster radius 1~ ~ for R, which thus
diverges with exponent ii. Thus his effective dimension d~is——

d~=y/v=d~~~.

(43)

The same result is also obtained by Forrest and Witten [138]from the correlation function; it wasalso supported by Mandelbrot in his earlier papers [136]but rejected in his past work [137].In two
dimensions, eq. (42a) gives an effective dimension of about 1.9, whereas Stanl~y’seq. (43) gives about
1.8. The difference increases with increasing dimensionality d; e.g. in si~dimensions we haved(p~)= 4 whereas d~(p~)= 2.
Finally, Kirkpatrick [7]defines an effective dimension d by requiring that foi~p —~Pc the mass withina region of linear extent ~increases as ~. Ifone looks at the infinite network fo~p slightly above p,, thenumber of network sites in such a region is about P..~cc p ~cc ~cd_~iv;thus d = d — 13/v =d/(i + 1/6) in this case, a result which agrees with d~(p~)
in eq. (42a). But Kirkpatrick [7]also looks atother quantities X which vanish at the percolation threshold as (P Pc)~The~reffective dimension is
= ci
f3~/
i’ and thus in general different from d = d(p~).For example, his backbone of the infinite
cluster has a f3~of about ~ in two dimensions, giving an effective dimension of the backbone [7]asabout 1.6.
We thus have seen that a variety of different definitions are possible, some of which lead to thesame result for the effective or fractal dimension. The choice of eq. (42a), firs~suggested by Harrison
et al. [109],———d’=d/(l+116)
seems to us particularly plausible for critical percolation clusters; moreover it is arrived at by the
majority of definitions and authors [7, 79, 109, 134, 137]. The relations between some of thesedefinitions are well understood in terms of percolation scaling and critical exponents. It seems likely[76]that similarity laws like eq. (41) will turn out to be relevant for a proof that the mathematicallywell defined [136]fractal dimension in the Hausdorff-Besicovich sense agrees with one of the effective
dimensionalities ci’ discussed here.
In conclusion, concepts like the fractal dimension of droplet-like ramified clusters are usefulcatchwords if one knows what they mean in the given context. But they ~an also be misleading
headlines when the real situation is somewhat complicated.
5. Lattice animals
So far we discussed properties of percolation clusters like the cluster numbers n3~p).These
2n, ~ jp — pcI~’,involving sums overnumbers
gave
us thisotherquantities
like themoments
L s 3 are mor~fundamental than theallclusterthen
sizes
s. In
sense
the cluster
numbersn, of section
moments like x derived from them. Now we go even one step further down to the basics and look at~

52

D. Stouffer, Scaling theory of percolation clusterseq. (7):n,(p)= ~ g,,p’(1 —p)’.

Obviously the number g~of geometrically different cluster configurations, which is independent of p,
is more fundamental than the p-dependent number of percolation clusters, since the latter can bederived from the former according to the above equation. Moreover, the number g~is simpler than
the number n,(p) since it does not involve the concept of probability. Purely geometric counting issufficient. In principle we thus reduced the whole percolation cluster problem to kindergarten
mathematics: How can be put together s square pieces of a domino game?
The number g,~is also called the number of (lattice) animals since it refers to the question: How
many different species of multicellular organisms can Nature form out of s single cells? (The word isdue to Harary [139].)Domb [1021reviewed the mathematical background on these animals. The firstscaling theory was attempted by Leath [73]whereas our scaling assumption here is taken from later
papers [71,89, 91, 126]. For animal numbers in subsection 5.1 we rely on refs. [42,45, 48, 93]; most of
the material in subsection 5.2 was not published before. We regret that in spite of the intrinsic
simplicity of the problem our description is more mathematical here than in the other sections.
Averages based on the gst alone are “animal” properties, whereas percolation quantities use g,5p’(lp)’ instead.
—

5.1. Scaling close to critical point
This part discusses the animal numbers g,~ for perimeter-to-size ratios t/s near and below thecritical point (1 pJ/pC; the following subsection 5.2 deals with those animals whose 1/s-ratio is near
the maximum of the perimeter distribution function for animals.
First one could think that g~1again equals, for large s and t near IC = s(l PC)/PC, a simple scaling
expression analogous to our other assumptions like eq. (15):——g,~cc ~

f~[(t—tC)5°1

But that assumption would be wrong as we will see now by deriving a presumably exact relation
between g, and n,(p), for all finite ratios I/s below (1 p~)/p~.
That result (45b) will be different fromthe above speculation. Readers may proceed directly to that eq. (45b) if they dislike our formulas in
between.
Throughout this section we denote the ratio 1/s by a and the quantity (1 p)/p by a,,. Thus
percolation clusters at the threshold have an average a near a~according to eq. (30a), and we may use
our previous scaling results to get animal properties for a near aC a,,~.Let us define
5e~fl,,(44a)g5, = A
where
A = A(a) = (a + i)~~/aa——

is a function introduced by Leath [73],andg =g(a)= iim[~ln(gs,A_3)]
is the exponent for the leading exponential variation of the ratio g~

5with size s, ref. [89].Thus the

1/A

D. Stouffer, Scaling theory ofpercolation clusters

53

remaining factor fl~in eq. (44a) varies weaker with size s than the other factors: lim,...[(l/s) ln fL~]
= 0 forall a. (These limits are takenat constant a = t/s and are assumed to exist and to depend on a only.)
For the numbers n~(p)of large percolation clusters we now getn5(p) =~

Ases~~,,ps(lp)’ =—f

exp{s[ln A + In p + a ln(1 p) + g(a)]}fTl,, dt.
—

(44b)

For large s the maximum of the integrand is determined by the brackets ~ since by definition thefactor Cl,, is less important. The expression In A(a) + In p + a ln(l p) has a maximum, as a functionof a, at ln(a + 1)— ln(a) + ln( I p) 0, i.e. at a = a,, (1 p)/p, as require4 also by eq. (30a). The
value of this expression at the maximum is zero, as the reader can easily ch~ckhimself. Thus if thefunction g(a) vanishes [891we can calculate the above integral by expaOding the bracket [..
quadratically in a a,,:2 [(a,,+ l~’—a’]= —~(a—a,,)2’ sI~.
lnA+ lnp + a~ln(l —p) =~(a a,,)Our result ~/s = a,,(a,, + 1) = (1 p)/p2 again agrees with eq. (29) for the width of the perimeter
—

—

—

—

—

—

distribution function. Eq. (44b) gives, with this assumption g(a) = 0,n,(p) =

J

exp[—~(a—a,,)2~~2/~]

Cl,, dlwith I = a,,s. From eq. (44a), with e’~= 1, we thus find
= n,(p)A’/(2ith~)”2
A = (a + l)a~Iaa; ~ = s(i —p)/p2

a = 1/s(45a)

provided p is calculated through t/s = (1 p)/p and the function g(a) vanishes. Similar to a Laplacetransform, we thus calculated g,, from n,(p).
When is this assumption g(a) = 0 correct on which eq. (45a) depends so much? We found above
than In A + ln p + a ln(1 p) + g(a) is zero at its maximum if g is zero. FOr nonzero g also thismaximum will have a finite value M Then the above evaluation of n,(p) will read to a factor esM forn,, apart from other factors varying weaker with s. Such a simple exponential decay of the cluster
numbers for large cluster sizes s is possible only below PC and not above PC [~0],as eq. (1 ib) shows.
Thus M must vanish for p above Pc, corresponding to zero g(a) for a below q~= (1 PC)/PC. (See ref.
[89]
for more discussion of this point.) Thus we found eq. (45a) to be valid above p,, i.e. for a <at,
but not below p,~,i.e. not for a > a~.For a below a~we find lim,....(g,,)U5 = A whereas in general wemerely have the inequality lim,....(g,,)”’ ~ A [88,89]. More discussion of lim(g~,)”~
is found in ref. [91]which has been critized in ref. [140].
For a above a~eq. (45a) cannot be valid exactly since fig. 12 has shown us already that for large
clusters the ratio I/s no longer equals, in the average, a,, = (1 p)/p. But for p close to p~,i.e. for aclose to a~,these deviations are quite small: lim,...(t,/s)—(1 —p)/p cc (PC~P)~’ [126],
as can bederived from eq. (31). Thus in the scaling regime of large clusters near the critical point we still may
try eq. (45a) on both sides of a~,at least for ~With n, cc s~f[(p —p~)s°]and ~, cc Vs we get [71,
—

—

—

—

73, 126]g,, ccAS . f(z)where
2(aC a)s~= p~(ac 4)sa.
z = (p _p~)sU= (a~ a)s°/(l+ a)(1 + a~)=(1 + aC)
5—r—1i2•

—

—

—

(45b)

54

D. Stouffer, Scaling theory of percolation clusters

Eq.(45b) is the scaling result for the animal numbers promised at the beginning. It differs by Leath’s

factor A’ from the speculation mentioned there. (Note that A’ can be rewritten in the symmetric formt~’/t’s’.)Direct numerical tests of eq. (45) are difficult [73] but it may look more reliable if one(1 + s)knows that it was first postulated [126]without this derivation and gave correct results for the
perimeter distribution function of percolation clusters, section 4.1.
Thus we derived a presumably exact expression (45a) for large animals with 0< 1/s <aC and ascaling assumption (45b) for I/s near aC. Only an explicit result for I/s much larger than a~is missing.
Since in a triangular lattice a = I/s can vary between zero and two for large clusters and since a,, = 1
there, one could first think that we solved more than half of the animal problem. But this is not so, as
one can see from fig. 19: Most of the animals have a ratio a = I/s near a
0 which is larger than a~.(Wecan see this effect already from fig. 12 where p = 0 corresponds to animals.) For example, in thesquare lattice we have a0 = lim,....(I,/s) = 1.2 for animals [52, 75] but a,, = lim,...(t,/s) = 0.7 for criticalpercolation clusters, eq. (30a). Thus for large s the difference between the average animal perimeterand the average percolation perimeter at Pc is about 0.5 s in the square lattice (and about 0.6 s in thetriangular, 0.3 s in 12.
the Thus
honeycomb
latticess [52]).
The width
the a~
perimeter
distributionfunction
for increasing
the regionof I/s of
below
covers only
an exponentially
increases
as fractions’decreasingonly
smallof the wings of the animal distribution. Therefore the next subsection 5.2
deals with the behavior near a0 where most animals live, not with the few domesticated animals neara~or below a,, which obey eq. (45).
Why is the animal problem a percolation problem at p = 0, as mentioned above [52,75]? (Note that
we keep s fixed and look at the average I, over all perimeters.) The animal perimeter I, is defined as
Y~,I g,t/~tg,,, the percolation perimeter as
I, = ~. n5t/~ n,~= ~I

g,,p’(l —~)‘/~
g,5p’(1 —p)’ =I

g,,(l —~)‘/~
g,~(1 —p)’according to eq. (28a). For p close to zero the factor (1 p)’ cancels out, provided various limits canbe interchanged. Indeed figs. 12 and 14 show that the limit p —*0 agrees with the result at p = 0, as
expected: At fixed size s the animals are the p —*0 limit of percolation clusters. In the method ofsection 2.2.3 thus simply all “animal” configuration were taken as equally probable by putting p = 0 in
the percolation computer simulation.
Figure 16 shows that theanalogously
densityanimals
are exponents
similar to [76].
eachTheother,D, (r) =I,7), profiles
to eq. of(41)large
but withdifferent
perimeters~°~“~°°‘°~
D4(,1s°~°~°°°
for animals obeys I, = a~s+ const according to series [52]and Monte Carlo [75]results, compatible
with the behavior expected for percolation clusters below p~,eq. (29).
—c—it/s

to

‘

as~i~

Fig. 19. Distribution of animal perimeters in the square lattice at s — 102. Critical percolation clusters have f/s near a
5 = (I — p5)/p, = 0.7 whereas

most animals have a larger perimeter with f/s near a0= 1.2, as can be seen from fig. 12. (Schematic.)

D. Stauffer, Scaling theory of percolation clusters

55

5.2. Exponents away from the critical point
To get the number of animals g,5 in the region 1/s a0 above a~we assume that the distributions are
2for large s:
Gaussian peaks with a width increasing as s~’= (c/2irs)”2exp[—~c(a a 2s] . g,(s..+co, a = t/s)
(46a)
—

0)

=

(46b)

g,,.

Here a0 = lim,.,..(t,/s) and c are suitable constants to be determined later. First we will deal with thetotal number g, of s-animals,2A(a~)’
eq. (46b).
at the critical point a = a~,eq. (45b), Sykes et al. [42,45]assume
to with
our g,,
s~~parameters 0 and A. Following a suggestion of Domb [102],Guttmann andg, ccSimilar. A’
twocc freeGaunt [93]fit the same series results more generally for large s by
cc

. A’

(46c)

exp(—const. s”)

with a third exponent w to be determined by “experiment” also. (w is related but not identical to the
exponent w in correction-to-scaling theories [87].)If w would be negative then the exponential factor
in eq. (46c) would be more important, for large s, than the prefactor s
cas~ingdoubt on the older
analysis [42,45]for 0 as well as on analogous assumptions like eq. (27) ~or percolation cluster
numbers. Fortunately a, turned out to be positive, whence we may expa~idexp(—const. ~05) =I const~s~’in eq. (46c). This correction factor is entirely analogous to eq. (~2)for n3(p~).Even the
numerical value obtained in two dimensions, w = 0.75 ±0.1, is consistent with ~hecorrection exponentof about 0.7 obtained for eq. (22) from Monte Carlo [261and series [19]data. Thus there is at present
no reason to doubt the validity of g, cc s
A’ for large enough s. The exponents 9 and w aresupposed to be universal, i.e. to depend on the dimensionality d only. F~rexample, 0(d = 2) =1.00 ±0.01 [93].For higher dimensions this exponent 0 was listed already in ref. [45,table II]; ford = 3 we have [93] w = 0.65 ±0.2. The parameter A depends on the lattice type; we have [93]A = 4.063±0.002, 5.183±0.001 and 8.34±0.02for the square, triangular, an~.lsimple cubic lattice,respectively.
In eq. (27) we have used the same exponent 0 for percolation clusters bek~wPc as for the animal
numbers in eq. (46c), since again the animal 0 is the p = 0 limit of the percolation 9(p): If the limits
p —*0, s —* ~ can be interchanged, then°,—n(p—*0)=~g,,p’(l—p)’=~g,,p’ cc s°(Ap)’.
Thus for sufficiently small p, the disturbing factor (1 p)’ cancels out, and the numbers g, of animals
become identical to the cluster numbers n,,n,(p —*0)= g,p’[cc s°(pA)’for ~ —*~]
(46d),
3. Therefore, for g, and n,(p —*0), also the two exponents 0 agree. Indeed,apartfrom analysis
a simple[49]of
factor pseries data [42]gave 0(P —*0) = 1.0 from the cIustei~numbers nnumerical
3 (p), in fullagreement with the exponent determined from the animal numbers for the s*me (triangular) lattice.
(See also ref. [80].)
Now we explain the value A = 4.06 in the square lattice, eq. (46c), by using our numericallydetermined percolation perimeters of fig. 12. Let a~= a,~(p)be the average perimeter-to-size ratio—

56

D. Stauffer, Scaling theory of percolation clusterslim(t,/s) for large clusters below the percolation threshold, where a.. = (1 p)/p no longer is valid. We
abbreviate lim,...((1/s) ln n,) by g1 = g1(p); this function vanishes for all p above p~according to eq.
(Iib). For large clusters, eq. (28c) gives a.. = (1 p)/p (1 p)dg,/dp, that means——

g1(p) = ln(plp,,) +

—

—

dp’.

On the other hand for p near zero we have, as mentioned above, n, cc s°(Ap)’, and thus g1(p —*0) =In A + In p + 0(p). In the comparison of these two results for g1 the diverging term log(p) cancels out,
and we get
lnA =ln(1/p,,)+

J

(47a)

~‘°~~‘~dp.

With the known data for a~~(p),fig. 12, and the numerical integration of eq. (47a) we get A = 4.1 ±0.1in
the square lattice, only one percent larger than the true result 4.06 from series approximations [42].
With similar methods we may even improve on the simple approximation (46a) to get the leading
s-dependence of g,~ for most ratios 1/5, not only near I/s = a0 and 1/s = a~,the “animal” and“percolation” averages. (As before we assume the limit of s and I going to infinity at constant ratio ato exist and to be a smooth function of a.) Equation (44b) tells us that a..(p) is determined by the
maximum of the function ln A + a ln(1 p) + g(a), with p fixed and a as variable. Since the aderivative vanishes at the maximum, we get dg/da = ln[(i + 1/a..)(1 p)] as the condition for a = a...
Integration gives——-g(a) =J—ln[(1 + 1/a..)(l —p(a..))] da..

(47b)

where again the relation between a.. and p e.g. from fig. 12 can be used. (It does not matter whether
the integration in eq. (47b) starts at zero or at a~since for all a.. below a~we have a.. = (1— p)/p, eq.
(30a). Thus the logarithm in eq. (47) and g(a) vanish there.) Figure 20 plots semiquantitatively thisfunction g(a) for the square lattice.
Equation (44a) gives lim,....((1/s) ln g,,) = In A + g(a) with a = I/s; and for a —* a0 we have from eqs.
(46, 47a) lim,.....((1/s) In g,~)= In A. Comparison gives, with the help of eq. (47b):
A = A(a0) e~0);

g(a0)

=J

ln[(1 + 1/a..) (1— p(a..))] da.

g (a

U08

0

-g

(47c)

I

0.8

~~1;
0.6

0.06

o.OL.
0.02
0

-

0.2

0.4

05

0.8

10

0.2

a

Fig. 20. Estimate of the (negative) function g(o) and its derivative, eq. (44a).

D. Stouffer, Scaling theory ofpercolation clusters

57

Since a0 = 1.2, fig. 12, and g(a0) = —0.1, fig. 20, for the square lattice we thus have
A = [(a0+~)ao+l/~ao}~~(ao)4.1,
again in excellent agreement with the desired result [42].We may also determine g(a0) from a0 and A,
as given by Duarte [52]for the triangular and square lattice, and again find ~(a0) = —0.1. We see noreason, however, that this quantity should be exactly universal. The numerieal agreement found for
A 4.1 in the square lattice seems to be the only test of this animal theory ki~ownat present.
(The constant c in eq. (46a) is given [89] by c = a~’(1+ a01’ g”(ao), which is about 2.4 in thesquare lattice. An unexplained exponent is the relation [48]s, cc t3~~2
for animals where the perimeter t
is fixed and one looks at the average size of animals which have a given perimeter,)Let us summarize our animal results here: For the leading s-dependence of the animal numbers g,,
we have a general result (for 0< I/s ~ ao)g,,—A’e’~ (A~(a+l)~~~/aa)(48a)with a = t/s, and a function g(a) given e.g. by fig. 20; g vanishes for a below a,, = (1 Pc)/Pc. Neara = ac, the region relevant for percolation scaling, also the prefactors neglected in eq. (48a) are
important and lead to eq. 2,,(a,,(45b):
a)s°].
(48b)
cc. A’ . f[pThe scaling function here is the same as that for the percolation cluster numbers in eq. (15), sincerather generally the cluster numbers and animal numbers are closely related by eq. (45a), for p above
p,,. In this sense the problem of most animal numbers seems solved in the asyihptotic limit: In general
the function g(a) in eq. (48a) describes the leading dependence on s; and the additional prefactors
needed for percolation scaling are given in eq. (48b).
=

—

—

—

~_T_U2

6. Other percolation problems
6.1. Random resistor networks
Already in subsection 1.1 we mentioned the question: how does an electric current flow through an
inhomogeneous medium? Imagine than each elementary cell or small cube Of a large simple-cubic
lattice is randomly either a piece of electrically conducting copper (with probability p) or a piece of
insulator (with probability 1 p). An electric current is supposed to flow from~one copper piece to its
neighbor only if both cubes have one surface in common; no electrical conbection is made by theedges or corners alone. Then an electric current flows through the whole lattice only if an infinite
network of copper pieces is present, i.e. for p above PC only. Let us for simplicity set the copper
conductivity equal to unity and the length of the lattice also equal to the unit length. The size of theelementary cube and the coherence length are then much smaller than unity. A unit voltage is applied
to two opposing faces (of unit area) of the lattice. The current now flowing through the lattice is called
the conductivity ~ = ~(p). We review this property only shortly here since our review concentrates on
percolation clusters. References [7, 13, 14, 15, 141—146] give real or computer experiments, and refs.
[7,146] review the scaling theory of conductivity.
Close to the percolation threshold p,~a positive exponent ~ (also called t) can be defined through
[3]
(49)
—

58

D. Stouffer, Scaling theory of percolation clusters

the conductivity is zero below p,,. Table 2 already listed numerical estimates for this exponent, butthey are in general less accurate than those for some other exponent. For example, two-dimensional
Monte Carlo data [7]give p. = 1.10 ±0.05 whereas a series prediction [108]is p. 1.43. Our estimate
p. 1.2 in table 2 is therefore not very accurate. How can we relate this new exponent p. to our earlier
exponents like o~and r?
Following Skal et al. [5, 141, 147] we imagine the infinite network slightly above p,, to consist mainlyof one-dimensional channels with a few crosslinks, fig. 21. The distance between two nodes is of theorder of the coherence length ~ cc I~
whereas the length of the chains connecting the nodes canbe longer and is assumed to diverge as (p p~Y”,with a new exponent ~‘ (in the literature our prime
is usually omitted). Of course this picture of one-dimensional channels is only an approximation [3]and requires that we remove all dead ends of chains which contribute to the percolation probability P..
but not to the conductivity I. Even then this “backbone” of the infinite network looks in reality[7,148] quite different from the above picture, fig. 21. Since modifications make it more complicatedwithout giving a simple generally accepted result [147]we still use here the simple picture of channels
to calculate ~, even though it may be wrong for d = 2.
Since the distance between two roughly parallel, current-carrying channels is of order ~, the total
number of such paths in our lattice is about l/~2 in three and l/~”’in d dimensions. Each path leads
from the top to the bottom of the lattice and thus consists of about l/~one-dimensional chainssections separated by nodes. Each section between two nodes has a length cc (p p,,~’;and thus awhole path leading from top to bottom has a length cc(pand a conductivity cc(p —p,,)’~.
Taking together all ~ cc (p _p,,)(d_I)~ such paths we find their combined conductivity to vary as
cc (p ~
Thus the conductivity exponent is [141]— p,,J_V———p,,)”~’(50)

It is tempting to assume [141]~‘= 1, i.e. p. = 1 + (d 2)v. Then we have p. = 1 exactly, apart frompossible logarithmic correction factors [149],in two dimensions. This result may be compatible withthe most recent computer simulations [7]and agrees with the epsilon expansion near six dimensions[108,147]. (In six dimensions we have [149]p. = 3 and ~‘ = 1.) But in two dimensions this assumption= 1 together with p
1.35 makes the above picture internally inconsistent: Then the length of thechain segments connecting two nodes at distance ~ would have to be smaller, not larger, than thisdistance ~if p is sufficiently close to p,~.Perhaps we have ~‘(d<4) 1 but ~‘(d>4) = 1 [108,149]. For
example, ~‘(d= 2) = v has been suggested [150].Or perhaps there is no simple relation between the
conductivity exponent p. and the cluster exponents like o and r. Contrary to our understanding of—I

H

Fig. 21. Approximation for the structure of the infinite current-carrying network of resistors. The true distance between the nodes is of order~ (P — PcY”, whereas the length of the chain segments connecting the nodes diverges as (p — pJ’, ~‘= 1, ref. [141].

D. Stauffer, Scaling theory of percolation clusters

59

cluster numbers and other cluster properties, no drastic improvement seems to have occured for thescaling theory of percolation conductivity since the older reviews [3]were written.
More fruitful were scaling ideas for relations between different conductivities if good and bad
conductors are mixed, instead of conductors and insulators. Then the cond~ictivityat p = p,~is not
zero; it increases smoothly with increasing concentration p of good conductots. Let RC0fld be the ratioof the conductivity of “bad” conductors to “good” conductors (set equal t~unity again). Now thetotal conductivity > depends on the two variables p Pc and RCOfld. Referenc~s[146,150—152] make ascaling assumption of the4 sameas we have seen it repeatedly in this review:
4[(p type
PC)(RCOfld)].
(Sla)
Y.(p, R,,00d) = (R,,Ofld)’
In this assumption analogous e.g. to eq. (15) a new exponent u and a new sealing function 4’ = 4(y)are introduced. For R,,Øfld = 0 we have again a mixture of conductors and insOlators and must recovereq. (49). Thus the scaling function 4i(y) varies for large arguments y as y5’, and then R,,
0fld cancels outof the conductivity in eq. (5la). (More precisely we constructed eq. (51a) sucth that R,,0fld cancels out
for R,,0~d= 0, as required by eq. (49).) Right at p = p,, it isM.thedifference
p —p~
p~for
which
has RCOto cancel
Finally,
below
small
out; thus 4i(0) has to remain finite, giving ~ cc (R~Ofld)
0d theconductivity must be proportional to R,,0fld since then the bad conductors, not~the good ones, give themain contribution to the resistivity. Thus 4’(y—*—~)cc(—y~~”,or cc RCOfld(pC—p)”~”. We now
summarize these three cases y—*+~,y—*O, and y—*—x:
cc (p p,,~
(RCOfldM(p = —*0 above p,,)
(Sib)
—

—

—

p~)

cc (R,,0fld)
cc (p,,—p)~~R~

0n~ (RCOfld—*0 below Pc).

(The positive exponent p./u p. is called s by Straley [1461;the exponent u is called s by Efros andShklovskii [151];and both papers denote our p. as t.) In the above discussiOn we took the “good”
conductivity as unity and the bad one as R,,Ofld; otherwise all results in eq. (51b) must be multiplied by
the “good” conductivity. If we mix a superconductor (infinite conductivity) ~ith a normal conductor,
this latter formulation has to be used; below p,, the conductivity then diverges as ~ p)~L_fLlU
multiplied with the normal (“bad”) conductivity arising from the factor RCOfld in eq. (Sib).
In two dimensions duality arguments [151,153] give u = ~ exactly, and thus p./u p. = p.: The
conductivity of a metal-insulator mixture film vanishes above Pc with the same exponent as theresistivity of a superconductor-metal film vanishes below Pc (Levinshtein [i2~]).In three dimensions
one has [146]u 0.7 and thus p.Iu p. 0.7 whereas u = 1 in six dimensions. Further exponents arelisted by Straley [146]and, for p. only, by our table 2.
Experimentally we mention the possibility [145]of varying p Pc very finely by using thedependence of the volume fraction p on temperature, if the two materials n~ixedtogether have twodifferent coefficients of thermal expansion; a reasonable exponent p. 1.8 was found in this way forthree dimensions. For more experiments see refs. [9, 143, 146]. Also the ~{alleffect in a resistor
network and the influence of random capacitors have been discussed [3, 154, 155].
—

—

—

—

6.2. Modifications of percolation

6.2.1. Miscellaneous
BOND PERCOLATION: What we discussed so far was mainly the site p~rcolationproblem withnearest-neighbor bonds. One can also look at “bond percolation” where each lattice site is occupied

60

D. Stouffer, Scaling theory of percolation clusters

and where each bond connecting two neighbors is randomly in one of two states. This bondpercolation problem seems to belong to the same universality class as site percolation, e.g. it isdescribed by the same exponents [105,107]. Also for the resistor network, the bond percolation
problem refers to mixtures of conducting and insulating connections between lattice sites. Actuallysome of the numerical exponents quoted earlier refer to the bond problem, not to site percolation. For
a combination of site and bond percolation we refer to ref. [156].In special cases [2],bond percolationis transformed exactly into site percolation in a different lattice.
LONG-RANGE INTERACTION: Most percolation studies define a cluster as a group of occupied
sites connected by nearest-neighbor distances. Again the universality concept asserts that the scaling
behavior (critical exponents; shape of the scaling functions) does not change if also next-nearest
neighbors are taken into account as parts of the same cluster, or if even longer ranges of interaction
are used. Work on the bcc lattice with nearest and next-nearest neighbors showed no peculiarities[26,69]compared with simpler lattices; and even longer ranges of interaction showed the same criticalexponents [24].
In the opposite direction goes a recent attempt to describe dilute magnetics where the local
magnetic moment is formed only if sufficiently many magnetic atoms are clustered together. Thus
each site in a cluster is required [157]to have at least m occupied sites as nearest neighbors. It is not
yet clear [157] if the scaling behavior of this model differs from the usual percolation problem(corresponding to m = 1).
POLYCHROMATIC PERCOLATION: If each lattice has not one but several states it can be in, thelattice now is no longer just black and white. Little is known [158]on cluster properties so far.
TREES ON A LATTICE: If in bond percolation on a lattice the clusters are restricted to be tree-like,with no cyclic bonds allowed (cyclomatic number = zero) then the scaling behavior is different fromordinary percolation. This problem corresponds [21,82] to the zero-state Potts model of section 2.3.2;
according to Wu [82]no phase transition occurs on the square lattice.
6.2.2. Continuous percolation
A major unsolved problem for percolation scaling is the question: How important is the existenceof a lattice structure? Are the exponents the same if the sites are distributed randomly in a continuum
instead of on a lattice? Computers like to have the sites on a square or simple cubic lattice, but many
random processes in nature happen in a continuum, not on a periodic structure. The problem isanalogous to a question in thermal phase transitions: Are the critical exponents calculated in a latticegas (Ising) model of fluids the same as those observed at real liquid-gas critical points of, say, H2O? In
the lattice gas the molecules are restricted, similar to lattice percolation, to sites of a periodic latticewhereas in real water they can move more freely. Recent high-precision measurements and improvements in the theoretical lattice-gas predictions seem to allow the conclusion that in three
dimensions the exponents are (nearly) independent of the lattice [159]. For percolation, the renormalization group approach also suggests the lattice to be unimportant. Monte Carlo cluster numbers ofFremlin [160] for continuum percolation seem not accurate enough to give critical exponents. We
refer to Webman et al. [161;see also 145] for the exponent p. of random resistor networks in acontinuum; earlier literature is cited there. More information on continuum percolation seems needed
to arrive at clear conclusions.

D. Stouffer, Scaling theory of percolation clusters

61

623 Interacting percolation

What happens if the probability of one site to be occupied depends on whether or not its neighbors
are occupied? A simple model to incorporate such correlations, which are ignored in random
percolation discussed so far, is the Ising model of ferromagnets (or lattice g~asmodel for fluids) at
finite temperatures T. Here probabilities are proportional to exp(— E/kB T) whei~eE is the energy of agiven configuration. In the Ising model we define
E=-J~~S~S1-H~S1

S1=±I.

(52)

The double sum runs over all pairs of nearest neighbors on a lattice. If the “spin” Si is positive we callthe site i occupied; it is empty for negative S1. The concentration p = ~(1+ t~S1))of occupied sites isvaried by the “field” H (chemical potential); the larger H is the larger is p. ((S1) is the average overmany configurations for one site i or, equivalently, the average over all sites excluding those
influenced by the boundaries of the system.) Clusters of occupied sites were investigated with thismodel in two and three dimensions, starting with refs. [33,35]. In two dimensiQns one has Pc = ~ [162with earlier refs.], which agrees with the “Curie point” at T~for magnetic properties.
(For H2n, diverges
for=H0 =the0“magnetization”(S1)
isnonzero
only
for
T
below
T,,.)
The
secondmoment~,sas (T — ~y 1.91 in two dimensions [163];this exponent is smaller than the corresponding
2.43 for random percolation, table 2, but larger than the exponent 1.75 for the magnetic
susceptibility d(S5 )/dH at H = 0.
In three dimensions, unfortunately the percolative and the magnetic phase transition happen at
different critical points, i.e. the threshold for interacting percolation is at a concentration p,,(T) smaller
than the concentration ~where the Curie point is located [35,162]. Thus if one wants to describe thethree-dimensional Curie point by a cluster model [34]
one has to define “clusters” differently: No
longer are they just groups of parallel spins with nearest-neighbor connections, as assumed in theMonte Carlo studies [35,37]. Instead they may represent fluctuations in an averaged local magnetization [36,96, 120, 127]. But little progress has been made in putting these ideas into a definition
simple enough for computer handling.
Recent progress in two dimensions makes the situation more hopeful there [25,115]. Since Curie
point and percolation threshold agree in two dimensions [162]
we may still defin~clusters as usual. At
H = 0 near T~the radius of typical percolation clusters diverges with the same exponent ii = 1 as thespatial range of magnetic correlations, ~ cc (T — T,,)”, ref. [25]. For fixed temperature T above T,, thetypical cluster radius diverges as (p,, p)”, with the same v 1.35 as for random percolation [25].
The cluster numbers at T = 2 T,, obey roughly a scaling relation like eq. (15); bi* the scaling function
f[(p pàs”] seems to have a shape different from that observed in random percplation [115].Thus, inspite of the results of Klein et al. [25] for the correlation exponent, the interacting percolation problem
does not share the same universality class with the random percolation problem, not even in two
dimensions.
Random percolation in this Ising model corresponds to infinite temperature T and infinite field H atfinite ratio HIT. For then the interaction between neighboring sites, which influences the occupation
probabilities through the ratio f/kB T, is negligible. Thus percolation can be regarded as both theinfinite- T and the zero- T!.Iimit of suitable magnets. (An even more complicat~dsystem [1641,themagnetic alloy, has on each lattice site one of two sorts of atoms; and each i~tomhas a magnetic
moment which points either up or down.) Monte Carlo studies of these Is~ngclusters at finite
temperatures are not only relevant for percolation but also for other questions~see ref. [165]for arecent paper with earlier references.
—

—

62

D. Stouffer, Scaling theory of percolation clusters

6.2.4. Dilute magnets at finite temperatures

The influence of finite temperatures is also a major problem for the dilute magnets of section 1.1,
which were discussed there only at T = 0. At nonzero temperatures thermal motion can break theexchange interaction between two magnetic moments within one cluster and make them antiparallel.
Then the methods of section 1.1 break down. Better theories ha-ye to distinguish between isotropic(Heisenberg) and anisotropic (Ising) magnets.
In the isotropic case one may look at “spin waves” as statistically independent thermal excitations;these “magnons” are harmonic waves in the direction of the local magnetization, just as sound waves
are harmonic waves in the local mass density. It turns out [3, 166] that the frequency of these spin
2, the exponents $ and p. for percolation
waves varies near p, as (p —p~)~’~/(wavelength)withprobability and conductivity, respectively. With this spin-wave approximation the variation of TC(p)near p,, and the dependence of thermodynamic properties on temperature for T —*0 has been
estimated. For example, T~(p)approaches T~(p~)= 0 roughly linearly in p. We refer to Shender [167]for details and earlier literature.
For anisotropic magnets the phase transition temperature T,,(p) vanishes near p~as 1/log(p — p,,).
The behavior near this trailsition is described by exponents which are in general neither those of thepure magnets nor those of percolation. We refer the .reader to Lubensky’s recent review [168].
7. ConclusIons

This review centered on the scaling theory of percolation clusters: How do the properties ofclusters depend on the cluster size s, and how does the phase transition at p, enter these properties?
The answers to these questions were described by several exponents o•, ‘r, 0, ~, p. Of these fiveexponents, u and r (and the critical exponents a, ~, y, 8, p derived from them) are defined only near
the percolation phase transition P = Pc, whereas the asymptotic behavior of cluster numbers and radiifor s —* ~ defines ~, 0, p both far away and near p~through eqs. (23, 27, 39b). We saw that the latter
exponents were different [70]above and below p~for example, ~‘(p<p~= 1 and ~(p > p~)= 1 — lidin d dimensions, eq. (26). On the other hand, the critical exponents o~an4 r are the same on both sidesof the phase transition and describe the cluster numbers there through eq. (15), a generalization of theFisher droplet model [34].Analogous expressions were proposed, as summarized in eq. (42), for theexcess perimeter, the radius, and the density profile of clusters. Such simple scaling assumptions arefamiliar from thermal phase transitions. We also saw two examples where scaling expressions like eq.
(42) are invalid: For the number of animals it is g 3, and not g
3jA
3j itself, which follows a scaling form,eq. (45b). And for the perimeter distribution, no scaling assumption like eq. (35) is valid. The
universality concept states that exponents and the shape of scaling functions do not depend on
microscopic details like lattice structure. We saw this assumption confirmed in general, but an
exception will be discussed in appendix 2.
In this sense the percolation clusters can serve as a simple introduction to scaling theory of criticalphenomena at phase transitions [12].For percolation the definition of the problem is particularly
simple. It requires little background knowledge like thermodynamics, magnetism etc., for percolation
merely combines geometry with probabilistic aspects. The more fundamental “animal” problem ofsection 5 even avoids the probabilistic concepts. Thus the animal problem, how to put together a given
number of dominoc squares, is defined on a kindergarten level. In high school we may ask for. theaverage number of percolation clusters obtained by flipping coins at concentration p = ~. Different

D. Stouffer, Scaling theory ofpercolation clusters

63

quantities are related to each other by scaling theory, which requires sometimes college-level
mathematics. More complicated “graduate” work is required to produce these quantities (like cluster
numbers on a computer, or exponents in renormalization theory). Our review skipped over thesecomplications but also pointed out unsolved problems.
In order to bring this review up-to-date we also covered problems, particularly in section 4, where
no consensus is evident from the present literature. Only the cluster number~of section 3 have been
confirmed by different authors with a variety of methods. Thus future research may not only put moreemphasis on points neglected here; it may also invalidate some conclusions particularly in section 4.
Figure 1 suggested already quite clearly that percolation is not yet solved entirely. As mentioned at
the beginning, the reader will get a more balanced view by reading other reviews, too [5—9].
We alsorefer to these reviews for a discussion of experiments, like the famous spread of disease in orchards[1—3].
Such experiments have seldomly given information on the cluster properties emphasized here.
Further research in the immediate future could center on testing scaling assumptions by Monte
Carlo work. The majority of present Monte Carlo studies is restricted to two dimensions. But three
dimensions are more relevant for applications, and higher dimensionalities are of theoretical interest
to study the transition to classical behavior at the phase transition. Continuum percolation is more
complicated but also more realistic than the lattice percolation emphasized here. A discussion ofcluster properties by renormalization group techniques has barely started [79,80,
114]. What cluster
radii do series methods predict? Of course, should an exact solution [25]of percolation become
available, such approximate methods would be only of historic and didactic value.
Even within the phenomenological scaling theory many questions are still open: Is the conductivity
exponent p. related to the cluster exponents o- and r? How can we explain the exponent 0 of eq. (27)below p~,and what is its true value above p~?Why is the radius exponent v0 below Pc close to ~in two
dimensions? What happens if clusters and animals are characterized by their perimeter t instead of
their size s [48],particularly near p = I? Which of the modifications of percolation mentioned shortly
in section 6 form a new universality class?
This review was a snapshot of the situation in December 1978, as seen by the author. Thus weconclude, for the reader’s amusement, with a historical remark. During tbe nineteen-thirties J.E.
Mayer was one of the founders of cluster theory for collective phenomena, as developed here in detailfor percolation near p~.In a review of his “Statistical Mechanics” textbook it was asserted quite
recently [169]that “the Mayer theory is no longer an active area of research in statistical mechanics”.
Mayer, in his reply [151],admits that the cluster development “certainly is obsolete in treatments ofcritical phenomena”. Throughout most of this review we explained critical phenomena by clusters.
Thus we hope to have convinced the reader that it is Mayer’s recent remark aild not his original ideas
which are “obsolete”. One merely has to select the right phase transition to study clusters: The
percolation problem.
We thank numerous researchers in the field for helping us throughout man~’years with preprints,
discussions and criticism. K. Binder, J.A.M.S. Duarte, H.J. Herrmann, W. Kin~el,S. Kirkpatrick, B.B.
Mandelbrot, H. Miiller-Krumbhaar, S. Redner, the referee, and in greatest detail P.J. Reynolds and H.
Nakanishi helped to improve earlier versions of this paper.

64

D. Stouffer, Scaling theory of percolation clusters

Appendix 1. Evaluation of sums
A.1.1. Derivations

How do we calculate the nonanalytic part of sums over all cluster sizes s, like [s,,s”ns]crjt in eq.
(16), if we assume the scaling assumption n~= q0s 7f(z), z = (p PC)SU to be valid, eq. (15). Basically
we replace the sum by an integral since only large clusters are responsible for singularities, andevaluate the integral apart from numerical factors by a substitution of variables. For this purpose wewill often use dz/ds = crz/s at fixed p.
For example, we find for the kth moment of the cluster size distribution:
—

~

=

J

5k~ ds = q0J

5k_~f[(p — Pc)S°] ds = (qolcr)f= qof36~p ~2”~ . q~3~
= I~ PcI

~

J

s

1+k—7

z~’f(z) dz. z~f(z)dz

J IzI~5~3.

f(z) Idz~.

(53)(In this appendix all integrals over z run from 0 to +~ above Pc and to
below.) Thus eq. (16) ~5
confirmed. (For the last equality we used the scaling laws (18).) Possible sign errors are easily
corrected at the end since both n~and f(z) are always positive. Thus we found the critical exponent
for the kth moment to be 2— a kfJ6 provided none of the following problems invalidates the above“derivation”: By replacing n. with q Tf(z) we made an error for small s, affecting additionalcorrection terms which are part of the0 sanalytic background. This background is unimportant if thewhole sum diverges or if only the “singular” part is investigated. Secondly, the prefactor qo may
depend smoothly on p, giving correction factors of the type I + O(p Pc). Again this error is not
relevant for the leading nonanalytic part of the sum. Finally, of course, the integral on eq. (53) must
exist, and here some caution is needed.
For large ±zthe scaling function f(z) decays exponentially, and thus for any finite k the integral ineq. (53) has no problems at ±Qc. But for z —*0 we normalized f(0) = 1, and the integral may diverge at
z = 0, which simply means we were not allowed to replace the sum by an integral right at the
beginning of eq. (53). In that case small clusters are more important than the typical clusters,s -~5E~For example, to evaluate ~,ii~we have k = 1 and jzI~8~”3= IzI~’,whence the integral
diverges at its lower boundary z = 0 for positive f3. The reason is simple: In L n, the main
contribution comes from s = 1,2, 3,..., and the sum has to give p(1 B(p — PC)~~ ~)according toeq. (4), if P.. = B(p Pc)~near Pc. Thus the nonanalytic part we are interested in is only a small
correction to the leading analytic background. Such problems occur whenever the sum in eq. (53) does
not diverge, i.e. when 2— a kf36 is not negative.
But even in these more complicated cases eq. (53) is still correct if we look at the nonanalytic part[~s skns]sing only. Why? In our above example, L,z~,we may look at the derivative with respect top. For ~ between zero and unity, as we assume from now on [50],this derivative equals —f3pB(,pPc)~+~.., and the analytic background term ~, . ns(pc) = PC has cancelled out. On the other hand,with dnjdp = qos°~f’(z),1’ = df/dz, we get
—~

—

—

—

—

—

—

D. Stouffer, Scaling theory of percolation clusters

J

qo

=

~

.

f’(z) ds = (qo/ff)=q01361p ~Pcf~’

f IzI(2_T~.

65

f’(z) jdzI

f z~- IdzI.
f’(z)(f’(O) is finite, and thus the integral converges at both boundaries.) The exponent thus agrees with theconclusion from eq. (53) with k = 1. Moreover, a comparison of the two expressions for the derivative
gives for the amplitude B above pç:

J

(54a)

B = —(q05/p~) z~ f’(z) dz.
-

Generally one may take the mth derivative (d/dp)” ~, skn If this sum contains a nonanalyticityp~~Cand one wants to determine the nonintegral positive exponent x, then with any m larger
than x this mth derivative diverges as I~ ~and can be evaluated as an integral:m ~ s5n, cc qo/38Ip _p~I(T_I_k_m~r)Iu ~fm~I+(l+k~r)/..
. f(m)(z) IdzI
(d/dp)
cc—f

cc I~_p~I_m~_l_~ff.

Thus we confirmed eq. (53) for the critical part of the sum: x = (r I k)/cr = 2— a kf3S. For
example, in the average number of clusters, k = 0, the third derivatives m = 3, finds out thenonanalytic part. (All derivatives ~(rn)= dmf/dzm exist if we assume as in section 3.1 that f(z) isanalytic in its argument.)
An alternative to this derivative trick is to subtract the disturbing background. Let us take again thesum L ps,, i.e. k = 1, and assume 0 <~3<1. Right at the percolation threshold we have~, - n,(p~)= Pc from eq. (4). Now we look above Pc at—

pP..=p—~s ~1Zs(P)~”Pc~

q

0

J

T[f(0) —f(z)] ds = qof3ôIp

J

B = (qof38/p~)—s n~(p)=~s[ns(pc)—n,(.p)]

s~This expression equals pB(ppercolation probability P..

—

pcf +-.-,

—

PcI~

J

lzI~~[f(0)—f(z)] IdzI.

(54b)

and thus we have found for the amplitude B of thez~’[f(0)—f(z)]dz

above Pc. The integral converges at the lower boundary since f(0) f(z) vanishes there as z. And,
triumph of mathematics, partial integration of eq. (54c) recovers eq. (54a).
For p below PC no infinite network is present, and L ~ n
5 (p) = p withou~the correction pB(pPc)~.Thus the integral in eq. (54b) must vanish below Pc, as pointed out (iii a similar context) by
Reatto [1341:
—

—

66

D. Stouffer, Scaling theory of percolation clustersJz~[f (0)—f(—z)] dz =0 orJ

z~f’(—z)dz =0.

(For clarity we replaced the integration variable z in eq. (54b) by —z in eq. (54d).) This “sum rule”[50,67] restricts the choice of the scaling function f(z). In particular f(z) cannot be symmetric aboutz = 0 since then the amplitude B would be the same on both sides of the phase transition. Equation(54d) also explains why there is a maximum in f(z) at negative z: The derivative f’(z) cannot have thesame sign for all z below Pc since then the integrals in eq. (54d) cannot vanish. The simplest choice forf(z) to obey the laws (54d) is to have a single maximum at negative z, and no maximum at positive z.
And this behavior is exactly what we observed in section 3.2. The Bethe lattice solution, with0</3 <1.
logf(z) cc —z2, is therefore a highly special case, incompatible with our assumption [50](Equation (54d) was challenged in ref. [89]but confirmed numerically in two dimensions [51].)
A. 1.2. Applications

With similar methods we evaluate the magnetization M = M(H, T) in the low temperature limit ofdilute magnets, eq. (5):M,

J IzI~’

tanh(IzI~. H,) . f(z) IdzI

M/~p p~I~= ±B+ (qof3t5/p~)
—

(54e)

with H, Hp./ksTIp ~Thus the “scaled magnetization” is a function of the “scaled field” only:
M, = M,(H,), and p Pc no longer appears in this relation explicitly. This scaling function M,(H,) ~5
given in eq. (54e) explicitly by an integral involving the cluster scaling function f(z). The amplitude B
for the spontaneous magnetization, B = M,(H, = 0), is the same as in eq. (54a, c) and vanishes on theparamagnetic side, eq. (54d). As required, only the ferromagnet has a spontaneous magnetization
M(H = 0) 0. In the ferromagnetic state, i.e. above Pc, the ±Bterm in eq. (54e) gives +B in thestable phase and —B in the metastable phase, if H >0. For negative H the signs are reversed. The ±B
term in eq. (54e) arises from the infinite network whereas the integral is due to the many finite
clusters. Of course, just as with nearly all other formulas in this review, eq. (54) is valid only for p
close to PcIn eq. (6d) the evaluation of ~, s - n,(pc) c_us is even simpler mathematically: We merely substitutey for hs and get
—

Pc

—

J

5 - fl5(p~)~
hs =2~qohT_J

S - n,(p~)(1— e_hs)

yl_T(1_e_Y)dy =q

J

0h~ ~

q0

-

~

I_T(l — e_hs) ds(1—e~~)dy
=p~E~
h~.

The amplitude E at p = Pc i5 therefore, after partial integration, given by
E = (qoS/p~)Jy~ e~dy = (q06/p~)I’(1 1/8)

where the gamma function of 1

—

—

(55)

1/8 is about 1.04 in two and about 1.16 in three dimensions. In thisD. Stauffer, Scaling theory ofpercolation clusters

67

way we calculated the prefactor q0 from the amplitude E [19,55]in section 3.2.1. Since E is usuallyclose to unity and 8 is quite large one may also get a reasonable estimate for q0 from Pc =Is 5 fls(Pc) -~q0 I, s~ = qo~(l+ 1/8) with the Riemann zeta function [170].
So far we introduced the amplitudes B and E. Also A and C may be defined by the total number ofclusters and the “susceptibility”, respectively (I2~pP..=_[~ s .n

[~
[~

fls]

sing

5]

=pAjp—p~j=pC~p—pcI~~2fl,]

—

[~.

sing

(56)

=pE - huo.

fls(pc)eh~]

sing

sing

Table 4 presents estimates for these four amplitudes; if two of them are known the others might becalculated by universality assumptions [50,55, 105].
All these complicated integrals are hiding somewhat the simple feature of our scaling theory: Only
one characteristic cluster size enters the calculations. For h = 0 near Pc this size is S~ IP Pct~
and when we work with e_~sat Pc then this characteristic cluster size can be taken as 1/h. Therefore
our sums can be evaluated by simple integrals. But even with more than one characteristic size, i.e.
when assumption (15) is violated, we may recover the scaling laws (18b) and similar results based onsums over all cluster sizes. For example let us slightly modify an ansatz of Reátto [134]and use as analternative to eq. (15) an ansatz with three free exponents u, r and x:
ns(p) = q0se_z - F,(z);Z = (p —Pc)S°
F, = [1+ const~exp(—(z + 1)Sx)]_I.
In the scaling limit of large clusters near Pc at fixed z we have F,(z) = 0 for z below 1, and F,(z) = I
for z above 1, similar to the Fermi function in quantum-statistical physics. (x and the constant are
positive.) Any sum over all cluster sizes can use, for its nonanalytic part, this ‘approximation of F,(z)being either zero or unity, and thus the third exponent x cancels out from t~iesesums. Neither theequation-of-state, eq. (20), not the scaling laws (18) depend on x. But this new exponent does enter the
shape of the cluster size dstribution below Pc and is not merely a correction-to~scalingexponent in the——

Table 4
Series estimates [19, 54—56] for the amplitudes A, B, C and E as defined ineq. (56), for various two- and three-dimensional lattices. (Abbreviations as intable 2). Both site percolation and bond percolation are listed. The dagger t
warns that a = - 2/3 was assumed [56];for the stars * Gaunt [55] gave
estimates assuming universality to be valid. For C the value below p,, is listed;
Monte Carlo results [26]give C(p > p~)/C(p> pj—’ 180 in two and 11 in three
dimensions (butsee also refs. [51,541).
B referstop above Pc, E top at p~,and A
happens in the calculated cases to be symmetric about p. [56].

Lattice:A (site)A (bond)
B (site)B (bond)C (site)C (bond)E (site)E (bond)

HC

SQ

TR

FCC

BCC

SC

D

1.53

1.53
1.55
0.147

1.56
1.60
0.128

4.2

*

*

*

*

*

*

*

0.101

0.142

0.185

0.26

0.134
1.09

0.084

0.041

0.074

0.122

0.222

1.lO

1.10

1.10

1.37
1.32

1.33
1.33

1.32
1.35

1.31
1.38

1.53
0.140
0.145
1.08
1.09

68

D. Stouffer, Scaling theory of percolation clusters

sense of eq. (22). If s—*~

below Pc, —z becomes very large, and log F5 then varies as (z + l)sx

—IzIs’~cc _~0±X•Thus below Pc (but not above Pc) the new exponent x enters crucially into the decay
exponent ~ = o• + x defined in eq. (23). In this sense two-exponent scaling, eq. (15), is violated for thecluster numbers n, but it is still valid for the sums over the cluster numbers. Fortunately we have seen
in section 3.2 that the additional assumption (15) is confirmed rather well: At present there is no needto introduce a third exponent for the cluster size distribution in the scaling limit.
So far we dealt only with the nonanalytic “singular” parts of the sums. If a sum diverges, itsleading singularity is also the singular part, and we have no problems in the scaling regime. But e.g.
I~n, and I~s n5 remain finite at Pc, with the nonanalytic parts appearing only as corrections
vanishing at Pc. These finite values at Pc cannot be calculated from the present scaling theory and are
expected, contrary to critical exponents etc., to be different for every different lattice structure. Such
non-diverging sums may lead to difficulties with averages. Let us assume that A. is any cluster
property (like radius R5) increasing faster than s ~ and slower than s ~ for I ~ s ~ s~and that it
obeys a scaling assumption similar to eq. (42). The average (A)~is defined by(A)re~ 5kflA/~

(57)

~

with a suitable k. (Polymer scientists call k = 0 the number average, k = 1 the weight average, andk = 2 the z-average.) For k = 2, 3,... both sums in eq. (57) diverge and thus get their main
contribution from s -~Sf; then (A) = A5~apart from numerical constants, and the average has theproperties which one would like it to have. But for k = 1 (and even worse for k = 0) there are surprises
hidden in the innocent eq. (57). For the denominator there remains finite (and equalto p~)indzthe
2_Tz_lf(z)Asccscaling
region,
and
only
the
numerator
diverges.
Thus
(A)
=
p~
s
n5A,
ds
cc
5
s
AscIp P~V~Thus a perhaps undesired factor I~ p~I”~
is the result of this definition of averages with
—

—

5000

k = 1.

The differences in the effective dimensionality of percolation clusters, section 4.3.3, are due to suchdifferences in definitions [106].Another example is provided by the labyrinthine ant [5, 171]. Ass’ ‘cthan an ant is parachuting into a two-dimensional lattice, filled randomly with sites below thepercolation threshold. (Experimental details on ant training were missing in ref. [171].)After landing
the ant runs away randomly but is restricted to the single cluster it has jumped into. After a long time
one measures how far the ant has run away from its landing site, and repeats the experiment manytimes. What is the average distance (R2) it has run away? Clearly that distance is connected with theaverage cluster radius. We leave it as an exercise to the reader to check which of the two expressions
in ref. [171], (R2) cc (p~_py2vand cc(p~_p)~2”
is the correct one for the type of average defined by
the jumping ant. In general we recommend to define averages through eq. (57) with k = 2, not withk = 1, since then they have nicer properties in the scaling region. For example, with A~= s we do notrecommend to call (s) with k = 1 a “mean cluster size”, as is often done. But with k = 2 eq. (57)
simply gives our typical cluster size sf for that average.
Appendix 2. One-dimensional percolation
In order to have at least one exactly solved problem in this review, which otherwise made no

attempt to be rigorous in its “proofs”, we now look at the special case of one dimension [112,113].
Percolation clusters in one dimension are chains of s occupied places; the two ends of the chain

69

D. Stauffer, Scaling theory ofpercolation clusters

border on empty sites. Thus the perimeter t is always 2, independent of cluster size. Therefore thecluster numbers are
2.
(58)
n
5(p)=p’(l—p)

For no p below unity can an infinite cluster exist, since already a single gap in the chain breaks it up
into two parts. At p = 1, of course, the whole lattice is one infinite cluster, in all dimensions. Thus wehave Pc = 1 for one dimension, and only the “paramagnetic” region below Pc is accessible. For P ~Pc
we rewrite eq. (58) with p = exp(ln p) = exp(p 1) = exp(p Pc) as——n
2 = ~_2[(pPc)S]2 exp[(p —pc)s] =5(p) = pS(p~_p)z (PPc)Sf(z)=z2ez.

Comparison with our basic assumption (15) shows that scaling is valid, with ~ = 1 and T = 2 exactly.
Equations (17, 18) then lead to the other exponents [113]a = 1, f3 = 0, y = 1, 6 = ~, v = 1, as alreadylisted in table 2. Unfortunately these exponents for d = 1 are not the limits for d 1, as table 2 shows,particularly for ‘y and v: The limits p Pc and d 1 cannot be interchanged.
A surprise is waiting [112]if the range of interaction is changed. Naive u4~iversalityassumptions,section 6.2.1, assert that then the exponents remain unchanged. But this is not so in one dimension[112].Let us define a cluster as a group of occupied sites connected by nearest or next-nearest
neighbor distances; thus our range of interaction is two lattice constants instead of one. Again we
have Pc = 1. Each cluster now has four empty neighbors as external perimeter~text = 4. But in addition
it may have tint internal perimeter sites, i.e. single holes of one empty site surrounded on both sides by
/s—l\
- ...
at least one occupied site. We have g5~=possibilities to distribute tint holes among the s
—*

—*

-~

~•

mt

)

—

bonds of an s-cluster. With a total perimeter of t = text + tint = 4 + ~ we thus get for the cluster
numbers:
s—I

n

t

/

4

1~

s

s—I

4

:5

15
~ ~+
=~
t~
;(1—p)”
tj..t=0\~int /
(p~— p)2s] = s 2z4z2)

5=~g51p(l—~)~ (2p) (Jp)t
4 exp[—
exp(—
(59a)
— (p~
p)
where z =(p—pc)s”2. (We used the binomial law and ln[(2—p)p]=ln[I---(l—p)2]=—(l—p)2=—(pc--p)2.)
Thus scaling is valid again, with r = 2, but now with rr =~ instead of unity: The—

universality assumption is violated because one of the exponents has changed. More generally, if
occupied sites separated by I lattice distances are still regarded as part of one cluster, one has [112]o~1/l,T=2, leadingto a 2—i, f30, ‘yl, 8=oc, ii=l.
This failure of universality in one-dimensional percolation can be hidden if instead of the scaling
variable —z = (Pc _p)~Uwe use the scaling variable z, ~(p. —p)’s°~
For the usull case I = 1 we now getnear Pc:n,(p) = s Tf(z,);
T = 2;
1(z
2 e~
(59b)
1) = z,with o- = 1. But in this form (59b) the scaling result obeys universality for general 1 and no longer
depends on the range of interaction, as one can see from eq. (59a) for 1 = 2, and from the results ofKlein et al. [112]in general.
One-dimensional Ising ferromagnets are similar to one-dimensional percolation in that they have aphase transition, at T = T~= 0, where only the paramagnetic side T> T~is ac~essible.Wiethege [172]looked at the specific heat of this magnet to find out if universality is violated also there, if longer

D. Stauffer, Scaling theory of percolation clusters

70

ranges of interaction are taken into account. If the energy needed to break up a bond between two
magnetic moments at distance 1 is J,, and if we define K1 I, 2J,/kBT, K2 1,1 - J,/kBT, we can expect
the specific heat to depend exponentially on K1 and K2 for T T~= 0. The calculation [172]showedthat for T near T~= 0 the specific heat varied as e_~’~1or e_K2, whatever was larger. Thus thetemperature enters the specific heat in a complicated way, depending strongly on the choice for theinteraction energy. Universality is violated much stronger than for percolation, where a simple
modification, eq. (59b), was sufficient to save the universality concept.
These two one-dimensional examples showed that simple and plausible concepts like universalityof critical phenomena need not always be true, even if they were confirmed for numerous otherexamples. This conclusion is regrettable but true.
—*

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73

74

D. Stouffer, Scaling theory ofpercolation clusters

Note added in proof. After completion of this review we learned of additional work of interest for thescaling theory of percolation. Some of these papers are listed below together with the section orreference for which they are most relevant:
1. Roussenq, C. Mitescu and H. Ottavi, Comptes Rendus Acad. Sc. Paris 288 B (1979) 49, on section 3.2 and ref. [23].
H.P. Peters, D. Stauffer, H.P. Hdlters and K. Loewenich, 2. Physik B, to be published, on ref. [75]in 3 dimensions, section 4.
S. Rednen, preprint, on radius of branching polymers [75, 133], section 4.2.2.
B. Rocksloh and D. Stauffer, preprint, on number of animals [48],section 5.2.
G. Shlifer, W. Klein, P.1. Reynolds and H.E. Stanley, pnepnint, on backbone exponents [7, 148], section 4.3.3.
J.P. Straley, prepnints, on ref. [82]and section 6.1.
P.G. De Gennes, 1. Physique (Paris) 40 (1979) L 197 on ref. [149]and section 6.1 and 4.2.2.
S. Redner and H.E. Stanley, prepnint, on theory for anisotropic percolation, section 6.2.1.
L.N. Smith and C.J. Lobb, Bull. Am. Phys. Soc. 24 (1979)360 and prepnint, on experiment for anisotropic percolation, section 6.2.1.
L. Turban, J. Phys. C 12(1979) L 191, on percolation with yoids, section 6.2.1.
L. Turban and P. Guilmin, 1. Phys. C 12(1979) 961, on Reich and Leath [157],section 6.2.1.
H. Nakanishi and P.J. Reynolds, prepnint, on site-bond percolation [156],section 6.2.1.
S.G. Whittington, G.M. Tonnie and D.S. Gaunt, J. Phys. A 12(1979) L 119, on Gaunt et al. [157],section 6.2.1.
Completed 9 May 1979.