Estimation of percolation thresholds via percolation in inhomogeneous mediaS. Zuyev and J. Quintanilla Citation: Journal of Mathematical Physics 44, 6040 (2003); doi: 10.1063/1.1624489 View online: http://dx.doi.org/10.1063/1.1624489 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/44/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in An introduction to inhomogeneous liquids, density functional theory, and the wetting transition Am. J. Phys. 82, 1119 (2014); 10.1119/1.4890823 Macroscopic uncertainty of the effective properties of random media and polycrystals J. Appl. Phys. 101, 023525 (2007); 10.1063/1.2426378 Numerical test of the Percus–Yevick approximation for continuum media of adhesive sphere model at percolationthreshold J. Chem. Phys. 114, 2304 (2001); 10.1063/1.1333681 Crossover behavior in the time evolution of the one-seed binary Eden model AIP Conf. Proc. 519, 371 (2000); 10.1063/1.1291589 Percolation for a model of statistically inhomogeneous random media J. Chem. Phys. 111, 5947 (1999); 10.1063/1.479890

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Estimation of percolation thresholds via percolationin inhomogeneous media

S. Zuyeva)

Department of Statistics and Modelling Science, University of Strathclyde,Glasgow G1 1XH, United Kingdom

J. Quintanillab)

Department of Mathematics, University of North Texas, Denton, Texas 76203-1430

~Received 23 January 2003; accepted 3 September 2003!

This paper mathematically justifies techniques used to estimate the percolationthresholds of fully penetrable disks, or Boolean models of planar disks. Generali-zations to systems of other particles in two or more dimensions are also discussed.© 2003 American Institute of Physics.@DOI: 10.1063/1.1624489#

I. INTRODUCTION

Accurate measurements of percolation phenomena are important in many areas of mathemati-cal physics.1–4 The Boolean model is a prototypical model for percolation studies. Recent esti-mates of the percolation threshold of the homogeneous Boolean model of random disks usedsimulations of nonhomogeneous Boolean models.5–8 In these articles, a nonhomogeneous Booleanmodel was simulated in a unit square, and the disks that were connected to the right edge of thesquare were found. The ‘‘edge’’ of these disks, called thefrontier, was found by using either thegap-traversal method or the more efficient frontier-walk method. Both methods saved computerresources by avoiding direct simulation of all disks within the unit square.

Using these methods, the critical density of disks of equal radiusR was found to belcrR2

50.359 072(4), where the parentheses represents the standard deviation for the last digit. This

density corresponds to a critical disk area ofacr512e2plcrR250.676 339(4).

These simulations were generalized8 to estimate the critical density for Boolean models withdisks of two different radii, where proportionf of the disks have radiusqR ~for 0,q,1) and theremaining disks have radiusR. The presence of disks with two different radii increased the criticaldisk area. The largest critical disk area simulated in Ref. 8 wasacr50.759 81(5); this was attainedusingq50.1 andf 50.99. Based on these simulations, it was conjectured thatacr is maximizednear f 512q2 for a fixed value ofq.

This paper provides a theoretical basis for the methods proposed in Refs. 5 and 6 for effec-tively estimating the percolation threshold of planar Boolean models of random disks. This isaccomplished by coupling homogeneous and nonhomogeneous Boolean models on the same prob-ability space. Because of this construction, classical results about homogeneous Boolean modelsmay be applied to nonhomogeneous Boolean models. Generalizations to higher dimensionalspaces and more general grains are possible; however, the proofs of these generalizations are quitetechnical and will be reported elsewhere.

Let C5@0,1#2 be a unit square with ‘‘left’’ sideL5$0%3@0,1# and ‘‘right’’ side R5$1%3@0,1#. Given a setS in R2, we will write tS5$ts:sPS% for the homothetical transform ofS, uSufor its area, and diamS for the diameter supxPSixi of S. Finally,b(x,r ) will stand for a closed diskof radiusr centered at a pointx.

We now formally describe the Boolean model of disks~or, more exactly, a family of models!.

a!Electronic mail: sergei@stams.strath.ac.uk; URL: http://www.stams.strath.ac.uk/;sergeib!Electronic mail: johnq@unt.edu; URL: http://www.math.unt.edu/;johnq

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 44, NUMBER 12 DECEMBER 2003

60400022-2488/2003/44(12)/6040/7/$20.00 © 2003 American Institute of Physics

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We will do this in a way which is suitable for our purposes; namely, we will couple a range ofmodels on the same probability space.

Let P̃ be a Poisson process in the phase spacex5R13R23R1 with intensity measuredt dx m(dr ), wherem(dr ) is a probability measure on the Borel subsets ofR15@0,̀ ). We willalso consider its subprocessP̃l , which is the restriction ofP̃ onto the set@0,l#3R23R1 . Foreach realization$(si ,xi ,r i)% of P̃, we define

Jl05 ø

~si ,xi ,r i !PP̃l

b~xi ,r i !,

which is a realization of thehomogeneous Boolean model of diskswith intensity ~density! l andradius distributionm for the disks. The disksb(xi ,r i) in the above union, denoted byKi for short,are calledgrains, and thexi are their correspondingcenters. By this construction,Jl

0 is anincreasing family of closed random disks, so thatJl1

0 ,Jl2

0 if l1,l2 .

This construction is equivalent to the usual definition of the Boolean model. The pointsxi—the projection ofP̃l onto R2—are distributed according to a homogeneous Poisson processPl with intensityl. Also, the radiir i of Ki are independently chosen for eachxi with probabilitydistributionm.

The critical percolation intensity for a homogeneous Boolean model is defined as

lcr5sup$l.0:P$Jl0 contains an infinite cluster%50%.

Another important critical value is

lN5sup$l.0:El0@Number of grains in the cluster containing 0 inJl

0#,`%.

HereEl0 is thePalm expectationwith respect toPl ~roughly, conditional expectation ‘‘givenPl

has a point at the origin 0’’!. For a large variety of percolation models, it has been proven thatlN5lcr . In particular, this equality holds for models of balls provided that the support of theradius distributionm is compact~see Refs. 9 and 10 and Ref. 11, Theorem 3.5!.

Along with Boolean modelsJl0, we will also consider their scaled versions, defined by

Jl0(t)5t21Jl

0. It is easy to see thatJl0(t) is distributionally equivalent to a Boolean model with

grainsb(xi ,t21r i), xiPP t2l . Obviously, given a scalet, the critical intensities forJl0(t) scale

correspondingly tot2lcr and t2lN .The estimation methods proposed in Refs. 5 and 6 are based on consideration of anonhomo-

geneousmodel a compact ‘‘window’’C. This modelJF(t) is the union ofb(xi ,t21r i) over(si ,xi ,r i)PP̃ such thatxiPC andsi<t2F(xi). In other words,JF(t) is a Boolean model inCwith balls of radiit21r i centered at the pointsxi of a nonhomogeneous Poisson processPF drivenby intensity measuret2F(x)dx. We assume that the functionF(x) is, in fact, a monotone functionof only the first coordinatex1 of x:F(x)5w(x1):@0,1#°R1 .

With this construction ofJF(t), the nonhomogeneous Boolean model has been coupled withhomogeneous Boolean models. Take anyxP@0,1#, and letl5w(x). Then all grains ofJF(t)

6041J. Math. Phys., Vol. 44, No. 12, December 2003 Estimation of percolation thresholds

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with xi<x are grains ofJl0(t), and all grains ofJl

0(t) with xi>x are grains ofJF(t). Thiscoupling will be used in the proofs of the following section.

Each horizontal line$y5c% may intersect the frontier in a few points. For everyc for whichthis intersection is nonempty, take the point with the smallest abscissa. The unionG t of such pointsis a subset of the frontier and consists of its points ‘‘visible from the left,’’ which we will call thecoastline. Actually, the mean first coordinate of the coastline points was used to estimate thepercolation threshold in Ref. 6, while the mean first coordinate of the frontier was used in Refs. 5,7, and 8.

Recall also that the Hausdorff distance between the two sets is defined by

r~A,B!5max$supaPA

infbPB

ia2bi , supbPB

infaPA

ia2bi%.

II. MAIN THEOREM

Theorem 1: Assume there exists0,R,` such thatm$(0,R#%51. Assume also thatw(0),lcr,w(1). Define pcr5 inf$p:w(p).lcr% and Q5@pcr,1#3@0,1#. If w is strictly increasing atthe point pcr , thenr t5r(Q t ,Q)→0 in distribution.

Proof: We need to show thatP$r t.e%→0 ast→`. From the definition,r t.e implies at leastone of the following cases:

FIG. 1. ~a! A realization ofJF(t) for t520 andw(x)5x. The setQ t of points connected toR is shaded. The thick blackline on the left face ofQ t is the frontierD t . The two vertical lines depictd1(t) andd2(t). ~b! As in ~a!, except witht5100.~c! A realization ofD t for t5400.~d! As in ~c!, except witht51000. Ast increases, bothd1(t) andd2(t) convergein distribution topcr .

6042 J. Math. Phys., Vol. 44, No. 12, December 2003 S. Zuyev and J. Quintanilla

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r t85 supxPQ t

r~x,Q!.e, ~1!

r t95 supxPQ

r~x,Q t!.e. ~2!

We will now show that both cases have vanishing probabilities ast grows.In case~1!, suchx cannot lie inQ, so there is necessarily a pointx of Q t with the first

coordinatex1,pcr2e. For t.8R/e, this implies the existence of a grainKt(yi)5def

b(yi ,t21r i)with pcr2e/2<yi

1<pcr2e/4 connected to @0,pcr2e#3@0,1# in JF(t)ù@pcr2e,pcr2e/4#3@0,1#.

Consider now the homogeneous Boolean modelJl1

0 (t) with l15w(pcr2e/4). By the cou-

pling construction described in the Introduction, all grains ofJF(t) with xi1<pcr2e/4 are also

grains ofJl1

0 (t). Therefore, the diameter ofJl1

0 (t) clusterWt(yi) containingKt(yi) is at least

e/4.DenotingC15@pcr2e/2,pcr2e/4#3@0,1# and using the scaling, we can write

P$r t8.e%<PH øyPPl1

~ t !ùC1

$diamWt~y!.e/4%J<ES (

yPPl1~ t !ùC1

1$diamWt~y!.e/4% D5ES (

yPPl1ùtC1

1$diamW~y!.te/4% D 5l1utC1uPl1

0 $diamW~0!.te/4%,

wherePl1

0 is the Palm probability with respect to the homogeneous processPl1. The last equality

is an application of the Campbell theorem~see, e.g., Ref. 13, p. 103!.The homogeneous modelJl1

0 (t) does not percolate sincel1,lcr , and so the diameter of its

clusters has exponential bounds. Applying Lemma 3.3 from Ref. 11, p. 68~see also Ref. 14!, wefind that

P$r t8.e%<A1l1et2 exp$2A2te% ~3!

for someA1 , A2.0. We see that limt→` P$r t8.e%50 for any e.0, implying that limt→` r t850 in distribution.

Let us turn to the second possibility~2!. Let n.2/e, and partitionQe5@pcr1e#3@0,1# inton blocks c(k)5@pcr1e,1#3@(k21)/n,k/n#, k51,...,n. If there is a LR crossing in eachc(k),then the vertical distances between adjacent crossings is always less than 2/n<e. Therefore, thereis no diskb(x,e) with a centerxPQ such that none of its points are connected toR by a path ofJF(t) insideC. In other words, case~2! doesnot hold for t if these LR crossings occur.

Let l25w(pcr1e). By the coupling argument, every LR crossing of the homogeneous Bool-ean modelJl2

0 (t) lying in Qe is also a LR crossing ofJF(t). By scaling and stationarity, the

probabilityp t of crossing such a block inJl2

0 equals the probability of a LR crossing of the block

@0 ,t(12pcr2e)#3@2t/(2n)0 ,t/(2n)# in the percolating modelJl2

0 . According to the classical

RSW theorem~see Corollary 4.1 in Ref. 11, p. 114!, p t tends to 1 ast grows to infinity. Insummary,

P$r t9.e%<12P$LR crossings for eachc~k! in JF~ t !%

<12P$LR crossings for eachc~k! in Jl2

0 ~ t !%512~12p t!n→0 as t→`.

h

6043J. Math. Phys., Vol. 44, No. 12, December 2003 Estimation of percolation thresholds

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Remark 1: A natural way to estimate the percolation threshold is to conduct a series ofindependent realizations of the Boolean modelJF(t) in a squareC for a range oft5t1 , t2 ,... .The exponential form of the estimate~3! shows that limn→` r tn

8 50 almost surely if the series

(ntn exp$2A2etn% converges~e.g., if tn5n). For the second case, the RSW theorem itself does notcontain an information about the speed at which the probabilityp t tends to 1, but the recent resultsfor discrete percolation suggest that it is at least exponential@see formula~8.98! in Ref. 3#. If true,the convergence ofr tn

in Theorem 1 is almost sure in the casetn5n.Recall thatD t andG t denote the frontier and coastline ofQ t , respectively. Not only doesQ t

converge toQ in the Hausdorff metric, but also the first coordinates ofD t andG t converge topcr .This is proved in the following theorem.

Theorem 2: Let pr1(D t) be the projection ofD t onto the first coordinate axis, and defined1(t)5 inf pr1(D t) and d2(t)5sup pr1(D t). Then both d1(t)→pcr and d2(t)→pcr as t→` or,equivalently, r(pr1(D t),$pcr%)→0 in distribution. AsG t#D t , then alsor(pr1(G t),$pcr%)→0 indistribution.

Equality ~4! is due to stationarity and isotropy of the modelJl3

0 (t), while ~5! is obtained after

scaling byt. By the RSW theorem,~5! converges to 1, thus concluding the proof. h

Remark 2:From simulations, it appears that the mean first coordinates of bothD t and G t

converge topcr at a rate proportional tot21, while the widthd2(t) –d1(t) converges to zero at arate proportional tot23/7.

III. GENERALIZATIONS AND SIMULATIONS

The proofs of the two previous theorems are based on two principal results for the homoge-neous Boolean models: in the subcritical region, exponential decay of the cluster diameter distri-bution and, in the supercritical region, the RSW theorem establishing the high probability ofcrossing of large blocks. The availability of these two results was the main reason for the condi-tions imposed on the Boolean model that were stated in the Introduction.

The proof of exponential bounds on the distribution’s tail can be generalized for Booleanmodels other than planar disks in all dimensions relatively easily by using the ‘‘spectral method’’proposed in Ref. 14. However, generalization of the RSW theorem demands substantial technicalwork. As inclusion of the corresponding proofs would go astray from the main subject of thispaper, they will be reported elsewhere. Here we just give the corresponding result without a proof.

6044 J. Math. Phys., Vol. 44, No. 12, December 2003 S. Zuyev and J. Quintanilla

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Let K0 be a locally compact separable space of compact setsK,Rd containing the origin andendowed with a suitables-algebra, and letm(dK) be a probability measure on it~see, e.g., Ref. 15for details on random compact sets!. A random elementKPK0 realized with distributionm iscalled atypical grain of a Boolean model. Consider a Poisson processP̃ in the phase spaceX5R13Rd3K0 with intensity measure ds dx m(dK) and its subprocessP̃F , which is the restric-tion of P̃ onto the set$(s,x):s<F(x)%3K0. Each realization$(si ,xi ,Ki)% of P̃ defines a randomset

JF5 ø~si ,xi ,Ki !PP̃F

~xi1Ki !,

which is a realization of thenonhomogeneous Boolean model. As above, we will also consider itsscaled versionJF(t), which is a Boolean model in a unit cubeC with grainst21Ki1yi centeredat the pointsyi of a nonhomogeneous Poisson process with intensity measuretdF(x)dx. Asbefore, we assume that the functionF(x) is a monotone function of only the first coordinatex1 ofx:F(x)5w(x1):@0,1#°R1 .

We make the following assumptions about the grain distribution of the Boolean model:

~A! finite radius: there existsR.0 such thatm$KPK0:diamK<R%51;~B! isotropy:m is rotation invariant;~C! nondegenerate connected grains:m$KPK0:uKu.0%.0 and m$KPK0:K is connected%51;

and~D! coincidence of percolation thresholds:lcr5lN .

FIG. 2. Realizations of inhomogeneous needles of lengtht21 in the unit circle.~a! In this figure,t510 and proportiona50.8 of them are oriented horizontally. None of the needles outside of the thick solid line are connected to the center, butmost of the interior needles are.~b! As in ~a!, except witht530. Only the needles connected to the center are drawn.~c!A magnification of the interior line fort5100; the needles are not shown. The interior pockets correspond to small areaswhere needles are not connected to the center.~d! As in ~c!, except fort5300. Ast increases, the frontier of the connectedneedles become circular, and the radius of this circle estimates the critical percolation density for the correspondinghomogeneous model. From this and other similar simulations, it appears that conditions~B! and~C! can be omitted withoutchanging the conclusion of Theorem 3.

6045J. Math. Phys., Vol. 44, No. 12, December 2003 Estimation of percolation thresholds

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Theorem 3: Assume the conditions (A)–(D) hold. Assume also that the functionw is such thatw(0),lcr,w(1) and strictly increasing at the point pcr . Then the statement of Theorem 1 holds.

A few remarks are in order. It was already mentioned in the Introduction that Condition~D!holds if all the grains are balls of a bounded radius~Ref. 11, Theorem 3.5!. It also holds for a largevariety of other percolation models: for example, grains which are star-shaped with the set of theirradius-vector functions being a compact in the spaceC(Sd21) of continuous functions on a spherewith sup-norm~see Refs. 9 and 10 for details!.

The example provided in Corollary 3.2 of Ref. 11, p. 52 shows that the condition~A! on finiteradius cannot, in general, be dropped without affecting the validity of condition~D!. If condition~A! does not hold, one cannot expect the procedure to be stable, as all ofC may be covered witha positive probability for all scalest.

Conditions~B! and~C!, in contrast, appear to be technical assumptions necessary for the proofbut unimportant for the estimation method to work. To see this, we consider the nonhomogeneousmodel in a unit ball~rather than in a squareC!. We also takeF(x) to be a radially symmetricfunction w(r ) with a pole in 0 and which decays to 0 whenr→1. Consider a range of Booleanmodels whose grains are segments of lengtht21 so that proportiona of them are orientedhorizontally and 12a vertically. As seen in Fig. 2, despite the anisotropy of the model and emptyvolume of the grains, the frontier of the cluster in a neighborhood of the origin has a circularshape. The radii of such frontiers estimate the critical percolation intensity for the correspondingmodels.

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6046 J. Math. Phys., Vol. 44, No. 12, December 2003 S. Zuyev and J. Quintanilla

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