Theoretical Physics

Percolation and Universality

Peter Full (900109-T199)full@kth.se

SA104X Degree Project in Engineering Physics, First LevelDepartment of Theoretical Physics

Royal Institute of Technology (KTH)Supervisor: Jack Lidmar

May 21, 2013

Abstract

In this thesis a detailed discussion of the topic percolation theory in squared lattices intwo dimensions will be conducted. To support this discussion numerical calculations willbe done. For the data analysis and simulations the Hoshen-Kopelman-Algorithm [2] willbe used. All concepts deduced will �nally lead to the determination of the conductance'sexponent t in random resistor networks. Using Derrida's transfer matrix program tocalculate the conductivity of random resistors in two and three dimensions [11] andthe �nite-size scaling approach were used. In two dimensions t/ν = 0.955 ± 0.006 wasobtained. Were ν is the exponent of the correlation length ξ in in�nite lattices. Thisvalue is in excellent agreement with Derrida (t/ν = 0.96±0.02, [11]) and slightly smallerthan Sahimi (t/ν = 0.9748± 0.001, [21]). In three dimensions the same approach yieldedt/ν = 2.155 ± 0.012 which some what smaller than the value found by Sahimi t/ν =2.27± 0.20 [21] and Gingold and Lobb t/ν = 2.276± 0.012 [25].

Contents

1 Introduction 2

1.1 What is percolation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 About the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background Material 5

2.1 De�nitions deduced from percolation in one dimension . . . . . . . . . . 52.2 Correlation length ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Fractal behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Power laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Scaling function for in�nite lattices . . . . . . . . . . . . . . . . . . . . . 82.6 Scaling function for �nite-size lattices . . . . . . . . . . . . . . . . . . . . 10

3 Investigation 11

3.1 Problem investigated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Used Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Tools for numerical calculations . . . . . . . . . . . . . . . . . . . . . . . 113.4 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4.1 Determination of the critical concentration for site percolation . . 123.4.2 Random Resistor Networks . . . . . . . . . . . . . . . . . . . . . 14

3.4.2.1 Analytical preparations . . . . . . . . . . . . . . . . . . 143.4.2.1.1 Conductance and its power law . . . . . . . . . 14

3.4.3 Determination of the conductance's exponent t . . . . . . . . . . . 15

4 Summary and Conclusions 22

Bibliography 24

1

Chapter 1

Introduction

1.1 What is percolation?

To get an insight in the concept of per-colation we will start the discussion with avivid example in nature. Assume we havea porous material such as a porous rock.What happens if we drizzle a liquid on topof an arbitrary piece of this rock? Will theliquid make its way through the piece ofrock or not? In case the liquid can �nd away from the top to the bottom one saysthat the system 'rock' is percolating. If thepiece of rock will percolate or not is deter-mined by the number and distribution ofholes in the rock. Several holes might forma tunnel (cluster) which can be use by theliquid to pass.

Figure 1.1: A porous rock was created for di�erentprobabilities p. The black pixel represent rock, thewhites holes in the rock and the blue ones are holes�lled with water that was drizzles on top of the rock.

Other examples are forest �res or Swiss-cheese. The forest �re will spread by ignit-ing trees close to an already burning tree.Reaching a situation where no more treescan be ignited the �re will extinct beforeit reaches the other side of the forest. If

there is always at least one neighbor treeto ignite the �re will make its way throughthe whole forest and reach the oppositeborder. The Swiss-cheese problem ques-tions if a slice of cheese will fall into severalpieces due to holes created during matu-ration. These three problems are not re-lated in nature but though share commonproperties. In all cases the properties ofa system depend on disturbance or �uctu-ations in the system itself. The amount,size, and distribution of these will e�ect thebehavior of the system. Many models ofpercolation are conceivable but this thesiswill only discuss a model which is mathe-matically described by a L × L × ... × Lobject where either the bonds (connectionsbetween lattice points) or the sites (latticepoints) are randomly occupied. Sites orbonds are occupied with the probability pand therefore empty with 1 − p. Impor-tant is that the sites or bonds are uncor-related randomly occupied. This ensuresthat an arbitrary site or bond will be occu-pied with the probability p independent ofany neighbor sites or bonds. Therefore thevivid examples from above cannot be calledcorrect by the de�nition used in this thesis.A rock might have cracks due to externalforces. This would cause holes which arenot uncorrelated distributed. However oneobserves properties in these systems which

2

Figure 1.2: Percolation patterns for di�erent probabilities p are shown in 250× 250 lattices for p = 0.3, 0.8and 500×500 for p = 0.59, 0.6. The biggest cluster is presented in red. By crossing the critical concentrationpc = 0.5927... a spanning cluster (connects the opposite edges) emerges.

are predicted by percolation theory. There-fore percolation theory remains a power-ful tool to make predictions about similarsystems in nature. Historically percolationtheory is attributable to Flory and Stock-mayer. During World War II they triedto describe polymerization. The expres-sion percolation theory was �rst used byBroadbent and Hammersley in 1957 whenthey introduced the concept of bond perco-lation.

Nowadays percolation theory is one ofthe best understood and well studied top-ics in statistical physics. Many geometricalproperties as the average cluster size, thesize of the biggest cluster, and fractal di-mension are known.

1.1.1 Phase transitions

Close to a critical concentration pc the sys-tems' behavior changes strikingly. Quan-tities of interest often follow power lawsand go to zero or in�nity close to the thecritical concentration. Therefore pc will oc-cur in almost all discussions. These strik-ing changes in behavior close to the criticalpoint are described by a second order phasetransition. A second order phase transi-tion is characterized by a diverging quan-tity close to the critical concentration pc. Incase of percolation theory one could namethe diverging correlation length ξ (see para-graph 2.2) or the cluster size of the biggest

cluster sbiggest (see Fig. 1.3).

Figure 1.3: A 200×200 lattices was investigated forvarying probability p. Data points were smoothed bygnuplot. Close to pc one observes the diverging sizeof the biggest cluster and the change in the probabil-ity to �nd a spanning cluster.

The behavior of a percolation lattice fordi�erent probabilities p is shown in Fig. 1.2.For p� pc many small clusters are formed.By approaching pc from below, less but big-ger clusters are formed. However no span-ning cluster exists. In an in�nite latticethe size of the biggest cluster sbiggest willdiverge at p = pc. In an in�nite latticeone �nds an in�nity cluster for p ≥ pc. Itwas proven that if we observe an in�nitecluster, it is unique [19]. For p � pc al-most the whole lattice is occupied by onebig, spanning cluster. In the following wewill use the expressions in�nite cluster or

3

spanning cluster, if we investigate in�nitelattices. In in�nite lattices these expres-sions are tantamount. For many investi-gations we have to use �nite lattices. Inthese cases we have to use the expressionspanning cluster for clusters that connectopposite edges of the lattice, since an �-nite lattice cannot contain in�nite clusters.Another property of phase transitions areorder parameters. These quantities are of-ten zero in one phase and unequal zero inthe other phase. In percolation theory onecould mention the strength P (p) of an in�-nite cluster or the conductance Σ(p). Thestrength P (p) is de�ned as the probabilitythat a site in an in�nite lattice belongs tothe in�nite cluster. For p < pc, P (p) iszero. For p ≥ pc, P (p) follows the powerlaw

P (p) ∝ (p− pc)β. (1.1)

The conductance Σ is de�ned as the recip-rocal of the electrical resistance. For theconductance Σ and p ≥ pc we will �nd

Σ ∝ (p− pc)t. (1.2)

We will study these quantities also for �nitelattices. They will then refer to the span-ning cluster instead of the in�nite cluster.Their proportional behavior will remain thesame although we have to take �nite-sizee�ects into account. For �nite lattices wewill �nd P (p) and Σ(p) unequal zero for pclose to pc but p < pc.

1.1.2 Universality

Universality is a concept that allows us tosubsume systems that are described by thesame critical exponents but di�er micro-scopically. Universality then will be usedto describe second order phase transitions.Therefore universality is only valid at the

critical point (in percolation theory pc).This is a powerful tool that is also validfor thermal phase transitions and used fre-quently in all kind of sciences.

The behavior of second order phasetransitions is determined by critical expo-nents. To name critical exponents onecould name the exponents that were men-tioned before in the power laws of P (p) andΣ(p). We obtain the the same critical ex-ponents for other two-dimensional latticesas the triangular lattice or the honeycomblattice. We get the same result for site orbond percolation. The critical concentra-tion pc might di�er for the di�erent latticesbut the behavior at pc is the same for alllattices with the same number of dimen-sions. Therefore we state that these lat-tices belong to the same universality class.The concept of universality was predictedby the renormalization group, which willnot be discussed any further in this the-sis. The behavior of systems at the criti-cal point in the same universality class aretherefore determined by only a few proper-ties as the dimension or symmetries but in-dependent of microscopic properties. Thiswill be useful in the following. From inves-tigations made for one type of system wecan make predictions for other systems inthe same universality class [4].

1.1.3 About the thesis

As orientation "Introduction to Percolationtheory" from Dietrich Stau�er and Am-non Aharony was used. Further readingabout basics and common quantities of per-colation theory an be done in Stau�er'sbook [1].

In the following site percolation in two-dimensional squared lattices will be thecenter of interest.

4

Chapter 2

Background Material

If not mentioned di�erent the contentof this chapter is created with the helpof Stau�er's "Introduction to PercolationTheory" [1].

As stated above percolation theory isone of the best understood topics in sta-tistical physics. A variety of systems hasbeen investigated. Many quantities in the-ses systems need to be calculated numer-ical while only some of them have exactsolutions such as the one-dimensional caseor the Bethe lattice, which are describedin Ref. [1]. A mathematical proof for thecritical concentration in a two dimensionalsquared lattice with site percolation wasgiven by H. Kesten in 1987 [5].

In the following fundamental conceptsof percolation and the relation betweenquantities of interest will be introduced.The chapter should furthermore give a bet-ter understanding of percolation theory ingeneral.

2.1 De�nitions deduced

from percolation in

one dimension

To start with percolation theory we will de-rive useful de�nitions exact solvable case inone dimension [1].

ns(p) is the number of clusters with ssites for the probability p in our lattice. For

a very long chain of lattices sites we �nd

ns = ps︸︷︷︸probability to �nd s

occupied lattice sites in a row

× (1− p)2︸ ︷︷ ︸probability forboth endpoints

.

(2.1)The probability to �nd a site that belongsto a cluster with the size s is nss. Since alloccupied sides belong to a cluster of certainsize the probability p equals

∞∑s=1

nss = p. (2.2)

The average cluster size S is de�ned as

S =

∑∞s nss

2∑∞s nss

. (2.3)

According to Eq. 2.2 the denominator willthen equal p. The numerator will betreated as followed

nss2 = (1−p)2

∞∑s

s2ps = (1−p)2(pd

dp

)2 ∞∑s

ps

(2.4)If we apply the geometric series for s =1 . . .∞ and p < 1 we get

S =(1 + p)

(1− p)(2.5)

In one dimension the critical concentrationis pc = 1. From Eq.2.5 we can see thatthe size of the clusters will diverge if p ap-proaches pc. This behavior was shown inFig. 1.3.

Now other important quantities in d di-mensions will be discussed.

5

2.2 Correlation length ξ

One of the most important quantity is thecorrelation length ξ. The correlation lengthis crucial for a system's behavior and ξ di-verges for p close to pc. Therefore we de-note

ξ ∝ |p− pc|−ν . (2.6)

In this thesis other quantities will be inves-tigated with respect to the scale that weapply on the system. We will observe dif-ferent behavior of these quantities depend-ing on the size of the chosen scale. Thescale is often compared with the ruler weuse to measure the system.

2.3 Fractal behavior

A fractal dimension Df is interesting forpercolation theory since it is also the expo-nent of power laws. For the following wewill denote D as the fractal dimension andd as the Euclidean dimension. A fractal isa mathematical object that has a fractal di-mensions. While familiar objects from ev-eryday life are spanned by Euclidean (inte-ger) dimensions (d = 2 for an area or d = 3for a volume) a fractal dimension D mightfall in between integers. Fractals are oftenself-similar patterns. While quantities inin familiar objects scale with the Euclideandimension d, the same quantity scales withthe fractal dimension D in fractals. To un-derstand this one could think of a piece ofpaper. If we double the size of this piecewe will also double the weight. The weightm scales therefore with m ∝ Ld, where Lis the length of the piece of paper. In frac-tals this proportionality is not longer valid.If we study the mass of the biggest clus-ter M(L) (number of occupied sites in thebiggest cluster) we will �nd the followingscaling relation M(L) ∝ LD. How can weunderstand this di�erent behavior? If wechose the scale to be very small we canmake a very detailed measurement. We will

register small holes or clusters and there-fore obtain a rather rami�ed pattern. Inthis case we would obtain fractal behavior.Whereas if we chose the scale to be large wewill ignore small clusters or holes in clus-ters and therefore obtain a homogeneouspatter as a sheet of paper is. In this casewe would obtain scaling with the Euclideandimension. The question arises below whatsize of the scale L we observe fractal behav-ior. In percolation theory the crucial quan-tity is the correlation length ξ. For scalessmaller ξ we observe fractal behavior andfor scales larger ξ we observe scaling withthe Euclidean dimension d. Therefore wecan denote for the asymptotic behavior

M(L) ∝{Ld, forL� ξLDf , forL� ξ

. (2.7)

The di�erent behavior for the density of thebiggest cluster ρ(L) is shown in Fig. 2.2.

Figure 2.2: The scaling of the density ρ(L, ξ) withL. For every probability 107 runs were made. Theinitial lattice site L′ was chosen to be L′ = 201 forthe triangular and the cross symbol and L′ = 151 forthe circle.

Therefore one can observe di�erent be-havior of the density for L > ξ and L <ξ. To visualize this e�ect a lattice withL′ = 201 was created with a probability pclose to pc. Subsequently it was checked if

6

Figure 2.1: The procedures of rescaling are presented. The left graphic shows a 1000× 1000 lattice at pc.One observes clusters of the size of the whole lattice. The top line illustrates a process of zooming in. The redframe represents one ninth of the current lattice and the section for the next graphic. Independent of the sizeof the lattice we observe a similar, rami�ed pattern. This illustrates self-similarity at di�erent length-scales.The bottom line shows coarse graining. In each step nine lattice sites where coalesced to one new site. Thatthese rescaling processes do not change the lattices properties is shown in Fig.2.3

the lattice contains a spanning cluster andif the lattice site in the center of the sam-ple was part of the spanning cluster. If so,a frame of size L = 3, was taken aroundthe occupied site in the center and the den-sity ρ(L) was determined. Then L was in-creased to obtain the next bigger latticewith L = 5 around the center. This proce-dure was repeated till the edge of the latticewith size L′ = 201 was reached. The resultis shown in Fig. 2.2. From the plot one ob-serves a change in the behavior of ρ(L) forL > ξ and L < ξ (ξ is shown by the dottedline on the x-axis). Unfortunately only fewdata points were measured for L < ξ but ifone measures the slope a of ρ(3) and ρ(5)one �nds a = −0.15 for the triangular andthe cross symbols. This is close to whatwe expected from the behavior of the massM(L) in Eq. 2.7. By dividing M(L) by L2

we get the density ρ(L)

ρ(L) ∝ LD−2 = L−0.104. (2.8)

It was used that D = 9148

= 1, 896. Theerror occurs due to the fact that we haveonly poor data for L < ξ. Using Eq. 2.6 wecan determine the correlation length's ex-ponent ν with the two correlation lengthsfrom our data. With ξ1 = 12 for (p− pc) =0.022 and ξ2 = 9.3 for (p − pc) = 0.035we �nd ν = 0.55 which is unfortunatelyfar away from the expected ν = 4/3. Thesame can be done with the exponent β ofthe strength P∞. P∞ is shown by the y-intercept. With Eq. 1.1 we �nd β = 0.14which is very close to β = 5

36= 0.139

from Ref.[21]. By comparing the calculatedvalues with what is expected from otherpapers one has to state that this methodis not suitable to determine quantities forpercolation theory. However the plot illus-trates the behavior of quantities for L < ξ

7

and L > ξ. So far we can make predictionsabout the asymptotic behavior of quanti-ties only. To study the behavior for L ≈ ξwe have to study scaling functions.

2.4 Power laws

If we study percolation systems at the per-colation threshold p = pc the correlationlength ξ will be in�nite. In this case allchosen scales L will be smaller ξ. We willobserve fractal behavior at all length scalesand we observe a self-similar pattern. As aresult properties of the lattice are indepen-dent of the chosen scale and the system isscale invariant. This requires the describ-ing formula to be independent of the met-ric. Self-similarity will allows us to rescalea lattice and subsequent blurry our eyesto �nd a system with the same propertiesas the initial one[1]. This e�ect is illus-trated in Fig. 2.1. Furthermore we expectcertain quantities of the system to followpower laws. Power laws are valid inde-pendent of the applied scale. In Fig. 2.3the distribution of cluster sizes at pc isshown. By using logarithmic binning (lat-tice sizes are not investigated separatelybut put into several bins covering the clus-ter sizes 1, 2−3, 4−7, ... in order to reducethe statistical �uctuation for large and rareclusters) one observes a straight line in thelogarithmic plot. This states that the prob-ability distribution of cluster sizes follows apower law at pc. To illustrate the scale in-variance the cluster size distribution of therescaled patterns in Fig. 2.1 were measuredand added to the same plot. Except for sta-tistical �uctuations (since only one latticewas investigated) we obtain the same prob-ability distribution. This con�rms that apower law is independent of the appliedscale and that the rescaling processes inFig. 2.1 do maintain the the properties ofthe initial cluster.

2.5 Scaling function for

in�nite lattices

As stated above the strength P has follow-ing properties in in�nite lattices: P = 0 forp < pc and P = 1 for p = 1. In between weassume a power law

P ∝ (p− pc)β (2.9)

for L � ξ. In the homogeneous regime(L � ξ) we can identify the density ρ ofoccupied sites belonging to the spanningcluster

ρ =M(L)

L2, (2.10)

with the strength P of the spanning clus-ter. This relation will be used later. Withthe discussion in paragraph 2.3 and Eq.2.9we know that

M(L) ∝{LD, L� ξ(p− pc)βLd, L� ξ

. (2.11)

If we are not at the percolation thresholdand thus ξ 6= ∞, we can choose L so thatit is in the order of ξ. It is a big assump-tion to connect the asymptotic behavior ofEq. 2.11 so that

const×LD = (p−pc)βLd, at L = ξ ∝ (p−pc)−ν .(2.12)

However this assumption will turn out tobe true later on. By substituting L withξ = (p− pc)−ν we get

(p− pc)−νD ∝ (p− pc)β(p− pc)−νd. (2.13)

This yields

D = d− β

ν. (2.14)

This relation is called the hyperscaling re-lation.

The density of occupied cluster sites isconstant for L � ξ. Therefore we can di-vide the whole cluster in several boxes withsize ξd without changing internal proper-ties. By doing so we obtain (L/ξ)d boxes.Since the boxes have the size of ξ theirmass scales as M ∝ ξD. Using this we can

8

Figure 2.3: The probabilityp(ns) to �nd a cluster of sizes is plotted against the sizeof the cluster s. The dotsshow the actual distribution fora 10000 × 10000 lattice in 13runs at pc. The crosses il-lustrate this distribution withlogarithmic binning. The or-ange symbols represent data ob-tained from coarse graining withthe factor b = 3 and the bluesymbols represent data from thezooming in in Fig.2.1. Thesquares are from the �rst rescal-ing step and the circles repre-sent the second step.

rewrite Eq. 2.11 for the asymptotic behav-ior

M(L) ∝{LD, L < ξξD(L/ξ)d, L > ξ

. (2.15)

Which can be con�rmed by

ξD−d = (p− pc)(D−d)(−ν)(2.14)= (p− pc)β = P

(2.16)

Regarding Eq. 2.15 we observe that the be-havior for the mass M(L) of the biggestcluster is less dependent on L or ξ sepa-rately but on the ratio L/ξ. Therefore itis natural to introduce a scaling functionwhich will be only dependent on one vari-able: the ratio L/ξ. Therefore we introducem(L/ξ)

M(L, ξ) = LDm(L/ξ), (2.17)

where the scaling function m(L/ξ) has toful�ll

m(x) =

{const, x� 1xd−D, x� 1

, (2.18)

to represent Eq. 2.15 for the asymptoticbehavior[1, 9].

One could have introduced the scalingfunction m(L/ξ) already in Eq. 2.11 in or-der to avoid the assumption about the be-havior ofM(L) for L ≈ ξ. This would haveyielded the same result.

Figure 2.4: The size of the biggest cluster M(L)against the lattice size L at the threshold pc. Theslope of this logarithmic plot is D = 1.9 the fractaldimension. This is what we expect for ξ =∞.

If we rearrange Eq.2.17 so that

M(L, ξ)

LD= m(L/ξ), (2.19)

substituting ξ with |p− pc|−ν and replacingthe mass of the biggest cluster M(L, ξ) bythe density of the biggest cluster ρ(L, ξ)Ld

we obtain

PLd−D = m(L |p− pc|ν). (2.20)

9

Figure 2.5: Thescaling functionfor the strengthP (L, ξ) = ρ(L, ξ)is shown withoutcorrection (left) andwith the correction(p − pc) (right).The same data asin Fig. 2.2 wasused. The slope isd−D ≈ 0.104

We use the strength P (L, ξ) since data forthis quantity was available. If we now plotPLd−D against L |p− pc|ν in a logarithmicdiagram we will reveal the scaling functionm(L/ξ). The result is shown in Fig. 2.6.To obtain the expected collapse of data fordi�erent L and p, (p − pc) had to be sub-tracted from the values on the y-axis, whichis shown in the right graph of Fig. 2.6. Oneobserves a constant value for the scalingfunction for L |p− pc|ν being small. Forincreasing L |p− pc|ν one observes a slopeof d − D ≈ 0.104. This is exactly whatwe expected from Eq. 2.18 and thereforesupports the introduction of a scaling func-tion.

2.6 Scaling function for

�nite-size lattices

So far only in�nite lattices were investi-gated but in all our applications we use �-nite cluster. With Eq.2.6 and 2.9 we �ndfor p > pc

P (p) ∝ (p− pc)β ∝ ξ−β/ν (2.21)

By introducing boxes of size ξd in the pre-vious section we already showed that someproperties of the in�nite scaling are trans-ferable to �nite lattices. More importantthan the actual size of the lattice was theratio between L and ξ. If we divide Eq. 2.15by the total volume Ld we will obtain thedensity ρ which is in the homogeneous dis-

trict de�ned as

ρ(L) =M(L)

Ld(2.22)

Dividing Eq. 2.15 by Ld will bring us thescaling function for the density dependingon L/ξ

ρ(L, ξ) ∝{LD−d, L� ξξD−d, L� ξ

. (2.23)

Applying Eq. 2.14 we get

ρ(L, ξ) ∝{L−β/ν , L� ξξ−β/ν , L� ξ

. (2.24)

Or alternative

ρ(L, ξ) = ξ−β/νx1(L/ξ), (2.25)

with the scaling function

x1(x) =

{x−β/ν , L� ξconst L� ξ

. (2.26)

This yields a general scaling law for anyquantity with the behavior X ∝ |p− pc|−χin in�nite lattices

X(L, ξ) = ξχ/νx1(L/ξ) ∝{ξχ/ν , L� ξLχ/ν , L� ξ

,

(2.27)with

x1(x) =

{xχ/ν , L� ξconst L� ξ

. (2.28)

This relation will be used for the followinginvestigation[1, 9].

10

Chapter 3

Investigation

3.1 Problem investigated

As stated above we will investigate randomresistor networks. This model is interest-ing for studies on alloys or similar mix-tures of two di�erent components but alsofor transport process and di�usion. In thismodel one component is a denoted to be agood conductor and the other one is a badone. Predictions about the conductance ofa sample can be made by applying percola-tion theory. The conductance will dependon the ratio between the components. Inpercolation theory the ration of the com-ponents will be determined by the proba-bility p that a site is occupied with a goodconductor.

3.2 Used Model

In order to do investigations on this prob-lem, squared lattices with bond or site per-colation will be studied. The focus now willbe on site percolation although the sameresearch could be done with bond perco-lation, as universality ensures. As usual,sites are randomly occupied with the prob-ability p. If two neighbor sites are occupiedthey will be connected with a good conduc-tor with a speci�c resistance R = Rgood.Otherwise the connecting is given by a badresistor with R = Rbad. To make the modeleasy to manage we will use in the followingRgood = 1 and Rbad =∞. Any other combi-

nation was conceivable but would not yieldfurther insight. A potential U will be ap-plied on two opposite edges of the lattice.The value of a current I = Σ× U throughthe lattice is expected to follow a powerlaw. Here Σ is the conductance, de�ned asthe reciprocal of the electrical resistance.

3.3 Tools for numerical

calculations

Although relations between di�erent quan-tities in percolation theory often can bedone analytically, many values of thesequantities have to be determined by nu-merical calculations. For this purpose pro-grams were written in c++. In order tomake numerical calculations, di�erent algo-rithms are needed to simulate and analyzerandomly occupied lattices. The Hoshen-Kopelman-Algorithm (HKA) o�ers a goodway to identify and distinguish di�erentclusters [2]. This algorithm is elaboratelyexplained in Ref. 3.2. Here only a brief dis-cussion will be done. A big advantage ofthe algorithm is that the time for the calcu-lations will be proportional to the numberof lattice sites. This o�ers a powerful toolfor big lattices. Another attempt found byM. E. J. Newman and R .M. Zi� which addsone site at a time and analyzes it immedi-ately afterwards [3]. Initially the randomnumber generator (RNG) rand() from the

11

c++ standard library was used. rand() isdenoted to be a bad RNG. Therefore theMersenne Twister from Ref. [22] was usedfor the further analysis. However the MTdid not yield better results for the studies.A comparison between these RNGs for theconductance is shown in Fig. 3.1.

Figure 3.1: The conductance for squared latticeswith lattice size L for the random number genera-tors rand() and Mersenne Twister(MT). The trans-fer matrix approach from 3.5 was used. The lengthof stripes was 107 for every data point.

The HKA will check if sites are occu-pied. If so the site will assigned to a clus-ter. If one of the neighbor sites is alreadyoccupied, the cluster label of this site willbe adopted by the actual site. If all neigh-bors are empty a new cluster label mα

s willbe created. For one cluster many clusterlabels might occur. This is the case if theconnection between two partial cluster isfound after each of the clusters receivedtheir own cluster label (see Fig. 3.2, lat-tice in the middle). The letter α is usedas the name of a cluster and introduced inorder to make the explanation of the HKAintelligible but α is not know during theanalysis. As mentioned above we can �ndseveral cluster labels for one cluster. Thisis called cluster multiple labeling techniqueand is one of the basic ideas of the HKA [2].We collect all the cluster labels belongingto the same cluster (all denoted by the in-

dex α) in one set of numbers

{mα1 ,m

α2 , ...,m

αs , ...,m

αt , ...} . (3.1)

The smallest of these cluster labels will becalled proper cluster label mα

t . To matchall other cluster labels to the proper clusterlabel another set of numbers will be intro-duced

{N(mα1 ), N(mα

2 ), ..., N(mαt ), ...} . (3.2)

Of this set of numbers N(mαt ) will be the

only positive number. Furthermore con-tains N(mα

t ) the number of occupied sitesin the whole cluster. All others are nega-tive and contain information about how tolink with mα

t by the following way

mαr = −N(mα

u), mαq = −N(mα

r ), ..., mαt = −N(mα

p )(3.3)

With this information the HKA can matchthe proper cluster number to all occupiedsites after a second run.

For the analysis of the collected thecommand-line program gnuplot (Version4.6 on windows, http://gnuplot.info/)will be used. In most cases power laws inthe form of

y(x) = b× x−α (3.4)

have to be studied. By using logarithmicscales for y and x we obtain

log(y(x)) = −αlog(x) +B. (3.5)

Were B = log(b). This visualization willyield a straight line of which the slope willdetermine the exponent α.

3.4 Numerical Analysis

3.4.1 Determination of the

critical concentration for

site percolation

The critical concentration pc is de�ned asthe lowest probability for which we expecta system to percolate. Therefore one can

12

Figure 3.2: A randomly occupied 7x7 lattice. (left): −1 represents an occupied site. −1 was chosen todistinguish from the positive cluster labels in the following procedure. (middle): After labeling the occupiedsites. Di�erent numbers represent di�erent clusters. In the bottom row one can see a cluster with index 3which is connected to a biggest cluster with index 2. (right): After the second loop all connected sites arelabeled with the proper label number.

Figure 3.3: The behavior of the probability pperc with p. The lines between the data points do not representactual data but are added by the function "smooth cspline" in gnuplot. This was done in order to to em-phasize the trend for di�erent lattice sizes L and to facilitate the distinction of di�erent series of experiment.The crossing point can be used to determine pc. Every data point was determined by the average of 105

repetitions.

research many percolation systems with in-creasing probability p (p is the probabil-ity that a site is occupied). By using theHKA we can check if occupied sites at bothends of the lattice share at least one com-mon cluster label. In that case we have acluster connecting both sides and the sys-tem is percolating. Plotting the proba-bility that the system is percolating ppercover the probability p that a single site isoccupied one can observe crititicality andcon�rm roughly the position of the criticalconcentration pc. More exact proceduresto determine the value for the critical con-centration pc have been used. An estab-lished way is the series method introducedby M.F. Sykes in 1964. This method will

not be use here but further reading can bedone in Ref. [7]. In the following systemswith varying number of lattice sites werecreated and check for percolation. Thisprocedure was done for probabilities p indi�erent intervals. For p < pc we expect alow probability pperc for the system to per-colate. For p > pc we expect pperc ≈ 1.Furthermore we should obtain a change inthe systems behavior close to the theoret-ical value pc,theo = 0.5927 [24]. Fig. 3.3shows the typical behavior of criticality.For p < pc we �nd pperc so be small andapproaches 0 as p departs from the criti-cal concentration. For p > pc, pperc ap-proaches 1. Furthermore one can see thatthe graph become sharper for bigger lat-

13

tices. This e�ect is caused by random er-ror and �nite size e�ects for smaller lat-tices with L 6= ∞. Especially the graphfor L = 10 in the second plot (red line)shows that for smaller lattices �uctuationsoccur, regardless to the high number of rep-etitions. For in�nite lattice one would ob-tain a sharp graph which jumps from 0 to 1at pc. This behavior is described by Kolo-mogrov's Zero-One law. It is applied in sta-tistical processes if one obtains for a out-come β with certainty β = 0 or for otherparameters β = 1 [6].

Our data is not accurate enough todetermine the critical concentration andas mentioned above there are other moreprecise approaches to determine pc. ButFig. 3.3 shows that all curves have one com-mon point in the vicinity of pc. How can weexplain that? We know for in�nite lattices

pperc =

{0 for p < pc1 for p ≥ pc

. (3.6)

For �nite lattices this trend was approx-imated for increasing L. We observe achange in behavior depending on the scaleL we apply on the system. Therefore it isreasonable to apply the scaling function wefound in paragraph 2.6.With m(L/ξ) = m(L/(p − pc)

−ν) =m(L(p− pc)ν) we get

pperc ∝ m(L(p− pc)ν). (3.7)

From this equation we �nd that pperc ∝m(0) for p = pc independent of L. Thisis con�rmed by Fig. 3.3, since all curvescross approximately the same point. Bydetermining the abscissa of the this pointwe can determine pc. From the plots weget pc ≈ 0.575 and pc ≈ 0.595. The secondresult which represents the plot with thelarger lattices di�ers only about 1% fromthe literature value of pc,lit = 0.5927 [24].However this value is not reliable. Not allcurves cross in one point. Furthermore onehas to take into account that the curvesdo not represent actual data points but are

added by gnuplot to emphasize the trendfor di�erent lattice sizes L. To get moreprecise values one had to investigate moreand larger lattices to minimize the �nite-size e�ect and run more repetitions to min-imize the random error.

3.4.2 Random Resistor Net-

works

3.4.2.1 Analytical preparations

3.4.2.1.1 Conductance and its

power law

This section will yield properties of per-colation beyond the purely geometric ones.As mentioned in the introduction percola-tion can be use in several application andact as model for many processes in nature.Random resistor networks represent an al-loy of two metals. One of the metals is aconductor, represented by occupied sites,and the other one is an insulator, repre-sented by unoccupied sites.

Center of interest will be the conduc-tance Σ. Σ is de�ned as the reciprocal ofthe electrical resistance R, therefore we willwork with I = ΣU .

In general it is important to distinguishbetween the conductance Σ and the con-ductivity σ. The conductivity is a propertyof the material whereas the conductance ita property of a device dependent on exter-nal parameter as the size or the tempera-ture. Since the conductance is inverse tothe resistance and thereby proportional tothe cross section w and and inverse pro-portional to the distance h, we will get thefollow relation between these quantities isΣ = w/hσ. The potential is applied onthe edges with distance h. Since we willinvestigate the conductance of squares inthe following, conductance Σ and conduc-tivity σ will be equivalent. Furthermore wedenote

Σ(p = 1) = 1. (3.8)

14

One can think of it as a squared sheetof graphite where squares are randomlypunched out with the probability 1− p. Ifthe system does not contain a cluster ofgraphite which connects two opposite edges(p < pc in an in�nite lattice) we �nd Σ = 0.From Eq. 3.8 we get that Σ equals unityfor a homogeneous sheet. The same behav-ior is know for the strength of the biggestcluster P . The question arises if these twoproperties are the same for pc < p < 1.Experiments by Lost and Thouless withgraphite paper in 1971 showed that this isnot the case.

Figure 3.4: A schematic sketch of the behavior ofP and the conductance Σ. In a random resistor net-works dead ends contribute to P but not to Σ.

The reason for that is that all sites ofthe spanning cluster contribute to P butnot necessarily to Σ since a multitude ofsites act as dead ends. Dead ends haveless than two independent connections tothe border of the cluster. The experi-ment reveals that Σ follows a power lawfor pc < p < 1. We denote

Σ ∝ (p− pc)t. (3.9)

With t being the exponent of the conduc-tance and not to be confused with the time.No analytically connection between t andthe geometrical exponents ν or β is known.As stated above, in in�nite lattices Σ fol-lows a power law for p > pc.

Σ ∝ (p− pc)t (3.10)

Using the result from chap. 3.11 we can in-troduce a scaling function for the conduc-tance

Σ(L, ξ) = ξ−t/νS(L/ξ) ∝{ξ−t/ν , L� ξL−t/ν , L� ξ

.

(3.11)

Thus Σ is independent of L for large L. Forsmaller L or at pc we expect

Σ ∝ L−t/ν , (3.12)

to be valid.

3.4.3 Determination of the

conductance's expo-

nent t

To determine the exponent t or ratherthe ration t/ν where ν is the correlationlength's exponent, one can use the transfermatrix program introduced by B. Derridaet al. in 1983 [10]. We use the model asdescribed in Chap. 3.2. To determine theexponent numerically we use a very longstripe which is shown in Fig. 3.5. Thestripe has a �xed width N and variablelength L. The stripe will be created foreach column during the numerical calcula-tion. The idea of Derrida was to apply apotential Ui between the bottom row N+1and each other row i = 1, 2, ..., N . The bot-tom row N + 1 will be grounded since onlythe di�erence in potential will cause a cur-rent Ii. For the calculations it will be con-venient to introduce a vector I containingall currents Ii and a vector U containing allpotentials Ui. We want to use with the sim-ple relation for scalars I = ΣU where Σ isagain the conductance (See Sec. 3.4.2.1.1).Now that we have introduced vectors for Iand U we have to introduce a matrix for theconductance. We chose the matrix AL inorder to avoid misunderstandings with thescalar conductance Σ or the sigma sign insums. The matrix AL is called transfer ma-trix. The index L represents the column ofour stripe we are investigating. The matrixAL will be modi�ed to AL+1 while creatinganother column and then store this infor-mation for the next column where it will berearranged. Thereby AL transfers informa-tion in between the di�erent steps of cre-ation. A big advantage is that AL will be

15

Figure 3.5: A very long stripe of �xed height N = 3 but variable length L is created recursive by addinghorizontal resistors hi and vertical vectors vi. The stripe is regarded to be semi-in�nite and therefore doesnot stop at the left border in the graphic.

adjusted for every run. Therefore we haveto store the information of one column onlyand not the whole stripe. This will enablestudies on huge lattices. The application oftransfer matrices is a common approach instatistical physics. By doing so we obtainthe vector equation

I = ALU, (3.13)

or

Ii =∑j

ALijUj, (3.14)

with i, j = 1, 2, ..., N + 1 representing theindex of di�erent rows. As one can seeall potentials are connected to any currentvia the transfer matrix. From Kirchho�'scurrent law based on the conservation ofcharge ∑

k

Ik = 0, (3.15)

we know that the amount of currents �ow-ing into the node is the same as �owing outof the node. This will required the trans-fer matrix to be symmetric. At the end ofthe numerical calculation we will be inter-ested in the value A11 (A00 in c++) sinceit represents the conductance between thetop and the bottom row by applying U1.

To start the calculations we chose an ar-bitrary column L. We assume that all sitesin column L− 1 are unoccupied. Only the

top and the bottom sites are always con-nected to other sites in their row (R = 0).Initially AL,initij = 0 for any i, j. To re-duce the e�ect of this assumption we willmake at least 105 runs where the stripe isextended by one column before we startthe actual measurement. The question willbe, how the transfer matrix A changes withfurther conductors and insulators. In eachrow we will start by inserting the horizon-tal resistors hi. The new potential U ′i willbe given by

U ′i = Ui + hiIi. (3.16)

Ui is �xed and hi the only connection be-tween the most right lattice site in row iand the grounded bottom row. If hi is aconductor (R = 1) U ′i will be reduced by Ii(since Ii is de�ned to �ow to the left). Ifhi is an insulator (R = ∞) the whole po-tential will deenergize at hi. We can writethis as matrix calculation

U ′ = U +HI. (3.17)

H is a diagonal matrix containing all hori-zontal resistors hi with

Hij = hiδij. (3.18)

The matrix AL+1 represents the conduc-tance in column L + 1. Therefore AL+1

is dependent on AL and the newly added

16

horizontal and vertical resistors hi and vi.To split the contribution of the horizontaland vertical resistors we introduce anothermatrix BL+1. This matrix will contain in-formation about the horizontal informationonly

I = BL+1U ′. (3.19)

By doing some conversions we will obtaina relation between AL and BL+1. We startwith Eq. 3.13 and 3.17

U = U ′ −HALU, (3.20)

which yields

U = (1 +HAL)−1U ′. (3.21)

With Eq. 3.13 and 3.19 we get

ALU = BL+1U ′, (3.22)

and by applying Eq. 3.21

BL+1 = AL(1 +HAL)−1. (3.23)

By adding a vertical resistors vi we con-nect two rows in column L + 1. Thereforea current ji might �ow between row i andi + 1 depending on the properties of theinserted resistor. We obtain

ji =[U ′i+1 − U ′i ]

vi. (3.24)

If vi is a conductor we �nd Ji = U ′i+1 − U ′i .For vi being a resistor we get Ji = 0. Thenew current I ′i due to U

′i will not only de-

pend on Ii but also on the additional verti-cal currents Ji−1 and Ji. Therefore we �ndfor I ′i

I ′i = Ii + Ji−1 − Ji

= Ii +[U ′i − U ′i−1]

vi−1−

[U ′i+1 − U ′i ]vi

= Ii +

[1

vi−1+

1

vi

]U ′i −

U ′i−1vi−1

−U ′i+1

vi,

(3.25)

with 1vo

= 0 for the top row. We intro-

duce another matrix V L+1 which will con-tain information about the contribution ofthe vertical resistors only

I ′ = I + V L+1U. (3.26)

V is tridiagonal. The elements are de�nedby

vij =

[1

vi+

1

vi−1

]δij−

[1

vi

]δji+1−

[1

vi−1

]δji−1

(3.27)If we now combine the contribution fromthe horizontal resistors BL+1 and the con-tribution from the vertical resistors V L+1

we will get the new AL+1 as

AL+1 = V L+1 +BL+1

= V L+1 + AL(1 +HAL)−1(3.28)

The transfer matrix AL+1 will contain theconductance for the stripe of height N andlength L. In two dimensions we will choseL = 107 in order to reduce the randomerror and the e�ect due to the boundaryconditions when we started the calculation.But we are interested in the conductancein squares. Therefore the obtained conduc-tance has to be divided by the length Land afterwards multiplied by the height N(this multiplication will not be necessaryin three dimensions). The conductance ΣN

per unit length will be given by

ΣN = limL→∞

AL11L. (3.29)

Using these equations we can determine theconductance for squares with di�erent sizeN numerically. By determining Σ for manydi�erent sizes N of lattice we can then de-termine the exponents t/ν [10, 11].

Important for our calculations will bethat we study the conductance at the per-colation threshold pc. For varying proba-bilities p the conductance depending on thelattice size N = 2, 3, ...300 was determined.The length of the stripes from the trans-fer matrix algorithm was 107 for all sizes.

17

The result is shown in Fig. 3.6. For p =pc = 0.592746 we observe except for statis-tical �uctuations a rather straight line. Ifwe choose p 6= pc the line of data pointsbends for growing N away from the datafor p = pc. This is shown in Fig. 3.6. Thisis what we expect from our model since forp > pc more sites are occupied with goodconductors than for p = pc which will yielda slightly higher conductance. This resultsupports our further investigations.

Figure 3.6: The conductance Σ for varying proba-bility p.

Using the transfer matrix algorithmof Ref. [11] one can show that the con-ductance Σ follows the power law inChap. 3.4.2.1.1

Σ ∝ N−t/ν . (3.30)

By introducing a constant a1 we get theequality

Σ = a1N−t/ν . (3.31)

The obtained data is shown in a logarith-mic plot. The curve will then be shown as

log (Σ) = − tνlog (N) + a1. (3.32)

The slope of data points is therefore givenby −t/ν. The constant a1 is so far not in-teresting. To minimize the statistical �uc-tuations within reasonable computing timeL = 107 was an appropriate length for the

semi-in�nite strip of conductors and resis-tors. Lattices with sizes between N = 2 toN = 300 were studied. The result is shownin Fig. 3.7. To determine the slope ofthe data points, a �t over all data points(dashed line) was done with gnuplot. Bydoing this we �nd t/ν = 0, 744 ± 0, 011.Obviously this line does not represent thedata points over a big interval. Further-more can be seen that your data picturesa slight right turn for increasing N . Sincecalculations were done with p = 0.592746which is very close to the acceptable thresh-old pc = 0.59274621(33) from Oliveira in2003 [24], another e�ect has to cause thebending in our data. So far we neglectedthe e�ects of �nite-size scaling. Eq. 3.31 isonly valid for in�nite lattices and a goodapproximation for N � 1. Since we an-alyze lattices from N = 2 on, we have totake further terms in account. Accordingto Ref. [21] the conductance scales with Nas

Σ(N) = N−t/ν(a1 + a2ln(N)−1 + a3N

−1) .(3.33)

This o�ers more parameters for the �t ofthe data. In Fig.3.8 the data was �ttedwith a function as in Eq. 3.33. Fits for allorders a1 6= 0, a2, a3 = 0 , a1, a2 6= 0, a3 = 0and a1, a2, a3 6= 0 were created. From theplot one can see that the best approxima-tion was made for all parameters unequalzero (solid line), despite that fact that thecontribution of a3 will decrease fast withincreasing N . For the zero order approx-imation (dashed line) we �nd as alreadymentioned above t/ν = 0, 744± 0, 011. Forthe �rst order approximation (dotted line)we obtain t/ν = 0.888± 0.006, a1 = 0.81±0.013, and a2 = 0.23 ± 0.001. This curveis closer to our data over a big interval butespecially for big N , where we expect to�nd the most reliable data, the discrepancybetween data and �t grows. For the sec-ond order approximation (solid line) we gett/ν = 0.949 ± 0.002, a1 = 0.880 ± 0.003,a2 = 0.214 ± 0.001, a3 = −1, 37 ± 0, 03.

18

Figure 3.7: The conductanceat p = pc for lattices in therange of N = 2 to N =300. The length L of the usedlattice strip was 107 over thewhole interval. The dashed line(t/ν = 0, 744 ± 0, 011) repre-sents a �t of all data point,the solid line (t/ν = 0, 954 ±0, 011)represents the �t for allpoint with N bigger than 50.

Figure 3.8: The data forthe two-dimensional caseat pc = 0.592746... fromabove was �tted with f(N) =N−t/ν

(a1 + a2ln(N)−1 + a3N

−1).For the straight dashed line a2and a3 were chosen to be zero.For the dotted line a3 waszero and for the sold line allparameters were unequal zero.

19

The second order approximation imitatesthe trend of the obtained data very good.That con�rms the qualitative correctionsdue to �nite-size scaling. With the de-termination of a1,2,3 we know the di�erencebetween the values expected form in�nite-size scaling and �nite-size scaling. The er-ror due to �nite-size scaling is

∆Σ = Σ∞ − Σfinite

= −N−t/ν(a2ln(N)−1 + a3N

−1) ,(3.34)

with a2 = 0.214 and a3 = −1, 37 from thesecond order approximation above. The er-ror ∆Σ was added to the data as error bars.The error bars represent the error due to�nite-size scaling only. The result is shownin Fig. 3.9. The end points of the errorbars can be seen as corrected data pointsand were used for a linear �t in the loga-rithmic diagram. To reduce the statistical

error only end points from N = 50 to N =300 were taken into account. This yieldedt/ν = 0.959± 0.005 and a1 = 0.91± 0.02.

Figure 3.9: The error bars represent the calcu-lated correction due to �nite-size scaling. The endpoints of the error bars were used to determinet/ν = 0.959± 0.005.

Figure 3.10: The conduc-tance Σ in three dimensionsat pc = 0.3117. The solidline represent a �t of f(N) =N−t/ν

(a1 + a2ln(N)−1 + a3N

−1).The dashed line is a linear �tthrough all data points. ForN > 20 the data reveals higherstatistical �uctuations. The reddot is excluded from the analysis.

20

The same procedure was done for thecase in three dimensions. Again Derrida'stransfer matrix approach was used. Thistime a semi-in�nite bar was investigated.The cross section of the bar is N ×N andthe height is the variable length L. Onedimensions has to be added to the algo-rithm explained above. Resistors are addedin the order: x-,y-,z- direction. For the xdirection the transfer matrix was modi�edas it was described in the two dimensionalcase for the horizontal resistors. For the yand z direction the procedure of the twodimensional vertical resistors has to be ap-plied. Otherwise the algorithm is the sameas described for two dimensions. The re-sult for the conductance Σ in three dimen-sions is shown in Fig.3.10. For N < 20 theheight of the bar was L = 107. For N > 20the height was reduced to L = 5 × 105

in order to work with feasible computa-tional time. This is shown in the higher�uctuation in data for N > 20. The dotshown in red (for N = 25) was excludedform the analysis since it deviates too muchfrom the trend of all other points. Fromthe linear �t over all data points we gett/ν = 1.73 ± 0.02. The exponent of thecorrelation length ξ in three dimensions isν = 7

8= 0.875(1) according to Ref. [20].

Also in three dimensions the linear �t overall data points does not match the datain big intervals. Therefore we apply the�nite-size correction. With a �t of the formf(N) = N−t/ν (a1 + a2ln(N)−1 + a3N

−1)one obtains t/ν = 2.127 ± 0.008 (using

the excluded data point one would havegot t/ν = 2.14 ± 0.02), a1 = 0.82 ± 0.1,a2 = −0.0007±0.02, and a3 = −0.73±0.06.Here it needs to be pointed out that the pa-rameters are de�ected with big errors. Fora2 we obtain an error of ≈ 3000%. This im-plies that the error of our result is biggerthan assumed from the following investiga-tion. Bigger values and longer bars (largerL) had to be investigated in order to reducethese errors. Using these parameter we canadd error bars. Again, the end points of theerror bar represent the position where weexpected the data to be in an in�nite lat-tice (N = ∞). From this linear regressionwe �nd t/ν = 2.103 ± 0.001. In contrastto the two-dimensional case the correctionof all data points were used for the linearregression.

Figure 3.11: The error bars represent the calculatedcorrection due to �nite-size scaling.

21

Chapter 4

Summary and Conclusions

LR FSC LR after FSCd ν t/ν t/ν t/ν2 4/3 0.95± 0.01 0.949± 0.002 0.959± 0.0053 7/8 1.73± 0.02 2.127± 0.008 2.103± 0.001

Table 4.1: The result from the studies of the conductance in 2 and 3 dimensions are presented. LR is theabbreviation for linear regression and FSC for �nite-size correction. In 3 dimensions all data points wereused for the analysis. In 2 dimensions only data with N > 50 was used for the LR.

By plotting the data obtained from the sim-ulations in a logarithmic diagram one ob-serves that the data points do not follow astraight line as expected from in�nite lat-tices but are bent especially for smaller L.This trend is described by the �nite-sizecorrections of the form

f(N) = N−t/ν(a1 + a2ln(N)−1 + a3N

−1) .(4.1)

That this formula with a1, a2, a3 6= 0 isa good approximation was shown duringthe discussion of the two-dimensional case.Fits using this formula match to the dataover big intervals. The same is true for in-vestigations in three dimensions. As ex-pected from the formula the correction forlarge N can be neglected. This was con-�rmed for the two-dimensional case. A lin-ear regression for N > 50 was made be-fore the �nite-size correction was appliedand afterwards. In Table 4 one observesthat t/ν of the linear regression before thecorrection is only about 1% lower thanthe value that we obtain afterwards. The

�nite-size correction however, seem to bea very good approximation over the wholeinterval. The value for t/ν is in excel-lent agreement with the the linear regres-sion for large N . If one corrects the datawith error bars using �nite-size correctionwe obtain a straight line for the endpointsof the error bar. The endpoints of the er-ror bars can therefore be seen as the datawe expected from studies of in�nite lat-tices. Taking the three di�erent values inaccount, t/ν = 0.955±0.006 is a good com-promise. This value is in good agreementwith Derrida (t/ν = 0.96±0.02) in Ref [11]and slightly smaller than Sahimi (t/ν =0.9748 ± 0.001) in Ref. [21]. Responsiblefor this discrepancy are probably the uncer-tainties that occurred during the �nite-sizecorrection. However the �nite-size approx-imation matches good with our data andtherefore cannot explain the whole discrep-ancy between our result and the ones madein earlier papers. To avoid errors due tothe �nite-size correction one could inves-

22

tigate larger lattices where the correctionbecomes negligible. The same observationsare valid for the three-dimensional case.The main di�erence occurred in the timeneeded to make the simulations. Since thecalculation in three dimensions are moretime-consuming only smaller lattices werestudied. This did not allow linear regres-sions for large values of N . The linear re-gression was done for all values which ex-plains the large discrepancy between thevalue for t/ν before and after the �nite-sizecorrection. However the values after thethe �nite-size correction in this work di�eronly about 3% from each other. Not tak-ing the value of the linear regression beforethe correction into account t/ν = 2.115 ±0.012 appears reasonable. Compared withSahimi t/ν = 2.27± 0.20 [21] and Gingoldand Lobb t/ν = 2.276±0.012 [25] our valueappears to be too small. This is expected

to be caused during the �t of the �nite-size correction. Some parameters were de-�ected with huge error (a2 = −0.0007±0.02which corresponds to ≈ 3000%). Takingthis into account our value is de�ected withhigher deviations than shown in Table4.In the papers of Sahimi and Gingold andLobb larger lattices (up to 803 lattices sites)were investigated. The values which areexpected to show the smallest mistake dueto �nite-size corrections are the ones withthe highest statistical �uctuation (in thiswork) since the length L of the bar hadto be reduced in order to maintain reason-able computational time. The length ofL = 107 seems to be su�cient to obtainreliable data. Therefore one could improvethe transfer matrix approach by investigat-ing larger clusters with a su�cient lengthof L in stead of using larger lengths L forsmall values of N .

23

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