Continuum percolation of polydisperse hyperspheres in infinite dimensions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Continuum percolation of polydisperse hyperspheres in infinite dimensions
Claudio Grimaldi
Laboratory of Physics of Complex Matter, Ecole Polytechnique F´ed´erale de Lausanne, Station 3, CH-1015 Lausanne, Switzerland (Dated:)We analyze the critical connectivity of systems of penetrable d -dimensional spheres having sizedistributions in terms of weighed random geometrical graphs, in which vertex coordinates correspondto random positions of the sphere centers and edges are formed between any two overlapping spheres.Edge weights naturally arise from the different radii of two overlapping spheres. For the case in whichthe spheres have bounded size distributions, we show that clusters of connected spheres are tree-likefor d → ∞ and they contain no closed loops. In this case we find that the mean cluster size divergesat the percolation threshold density η c → − d , independently of the particular size distribution. Wealso show that the mean number of overlaps for a particle at criticality z c is smaller than unity,while z c → − d for d → ∞ . I. INTRODUCTION
Percolation phenomena are ubiquitous in many aspectsof natural, technological, and social sciences, and theyarise when system-spanning clusters or components of,in some sense, connected objects form [1, 2]. A quan-tity of much interest is the percolation threshold, whichmarks the transition between the phase in which a gi-ant component exists and the one in which it does not.In general, the percolation threshold is a nonuniversalquantity, as it depends on the connectivity properties ofthe specific system under consideration [3]. For example,in continuum percolation systems, where objects occupypositions in a continuous space, the threshold dependson the shape of the objects [4–8], on their interactions[9–11], as well as on the connectedness criteria [12, 13].In this article, we consider the infinite-dimensionallimit of a paradigmatic example of continuum perco-lation: the Boolean-Poisson model [14, 15]. In thismodel, penetrable spheres with distributed radii havecenters generated by a point Poisson process, and anytwo spheres are considered connected if they overlap. Fora given distribution of the radii, the percolation thresh-old is given by the critical concentration η c of spheres, orby the critical volume fraction φ c = 1 − e − η c , such thata giant component of connected spheres first forms. Pre-cise numerical estimates of η c have been obtained in twoand three dimensions for, respectively, disks and sphereswith fixed or distributed radii [16–23]. The general trendobserved by these investigations is that η c depends on theform of the distribution function of the radii, and that ithas its minimum when the sphere radii are monodisperse(i.e., when the spheres have identical size). This lastpoint has been formally confirmed in Ref. [24], althoughit may not hold true in the limit of infinite dimensions d [25, 26].Here we show that for bounded distributions of theradii, that is for polydisperse spheres with a maximumfinite value of the radius, the percolation threshold of the Boolean-Poisson model tends asymptotically to a univer-sal constant as d → ∞ , provided that the radii distribu-tion is independent of d . This constant coincides withthe value found in Refs. [27, 28] for spheres of identicalradii, η c → − d , and it is independent of the particu-lar form of the size distribution function. We interpretthe universality of η c as being due to the statistical ir-relevance of the spheres with smaller radii: the onset ofpercolation is established effectively only by the subsetof spheres with maximum radius. Furthermore, we showthat the mean number of connected spheres per particleat percolation, z c , is less than unity for polydisperse dis-tributions of the radii, while z c → η c for d → ∞ is not universal, as it depends on theparameters of the distribution, and that it can be smallerthan the critical threshold of monodisperse spheres, incontrast to what is expected for finite dimensions [24]. II. THE MODEL
To construct the Boolean model, we consider N pointsplaced independently and uniformly at random in a d -dimensional volume V . Each point is the center of asphere with the radius drawn independently and ran-domly from a given probability distribution function ρ ( R ). If we denote N the number of spheres of radius R , N the number of spheres of radius R , and so on, FIG. 1. (Color online) Connectedness criterion for sphereswith different radii. (a) The spheres of radii R and R overlap the sphere of radius R , forming links between R and R and between R and R . (b) Corresponding clusterformed by nodes (sphere centers) labeled by the sphere radiiand weighted links (solid lines) connecting the nodes. we can write the following without loss of generality: ρ ( R ) = X i x i δ ( R − R i ) , (1)where x i = N i /N with i = 1 , , . . . is the fraction ofspheres of radius R i .Given any two spheres of radii, say, R i and R j , weassign a link between their centers if the spheres overlap,that is, if the distance r between their center is smallerthan R i + R j , as shown in Fig. 1. We express this criterionfor the formation of a link in terms of the connectednessfunction: f ij ( r ) = θ ( R i + R j − r ) , (2)where θ ( x ) = 1 for x ≥ θ ( x ) = 0 for x < R i centered at the origin. Theprobability that a second sphere of radius R j forms a linkwith the first sphere is: v ij ex = 1 V Z d r f ij ( r ) = Ω d ( R i + R j ) d V , (3)where d r is an infinitesimal d -dimensional volume el-ement at the position r of the sphere of radius R j ,Ω d = π d/ / Γ(1 + d/
2) is the volume of a sphere of unitradius, and Γ is the gamma function. We note that v ij ex defines also the excluded volume V ij ex = Ω d ( R i + R j ) d inunits of V between two spheres of different radii. III. IRRELEVANCE OF CLOSED LOOPS FOR d → ∞ An important aspect of the topology of random geo-metric graphs is represented by closed loops (or cycles) of connected nodes. The most studied loop quantity isthe three-nodes cycle c (3) d , often denoted the cluster coef-ficient, which gives the conditional probability that twonodes are connected given that both nodes are connectedto a third one. c (3) d has been calculated for systems ofspheres with identical radii and for any dimensionality[28, 30]. The observation that c (3) d vanishes exponen-tially as d → ∞ indicates that random geometric graphsin large dimensions have a locally tree-like structure.Using results from the theory of hard-sphere fluids, itis actually possible to show that, in the limit of large di-mensions, closed loops are negligible also for any numberof nodes and for bounded radii distributions. Randomand weighted random geometric graphs have thus tree-like structures when d → ∞ . To see this, let us firstconsider the case of monodisperse spheres with radius R M . We define an n -chain graph as a cluster of n ≥ n − n -chain that each haveonly one edge. The n -cycle coefficient c ( n ) d is defined asthe conditional probability that two nodes are connectedgiven that they are the end-nodes of an n -chain. Sincethe spheres have identical radii, we omit the subscriptsin Eq. (2), and we write the connectedness function assimply f ( r ) = θ (2 R M − r ). From the definition of c ( n ) d ,we can thus write: c ( n ) d = R dr ( n ) f ( | r − r | ) f ( | r − r | ) · · · f ( | r n − r | ) R dr ( n ) f ( | r − r | ) f ( | r − r | ) · · · f ( | r n − − r n | ) , (4)where dr ( n ) = d r d r · · · d r n . Besides a prefactor, theabove expression coincides with the cluster integral ofa ring of n hard-spheres of radius R M [31], as theMayer function f M ( r ) for a fluid of hard-spheres is just f M ( r ) = − f ( r ) [3, 32]. To evaluate Eq. (4) for d → ∞ ,we thus use known results from the theory of hard-spherefluids in infinite dimensions. Noting that the denomina-tor of Eq. (4) (i.e., the n -chain contribution) is simply V V n − [33], where V ex = Ω d d R dM is the excluded vol-ume for spheres of identical radius R M , and introducingthe Fourier transform ˆ f ( q ) of the connectedness functionwe rewrite Eq. (4) as: c ( n ) d = 1 V n − Z d q (2 π ) d ˆ f ( q ) n . (5)The integration in Eq. (5) for d → ∞ has been workedout in Ref. [31] (see also Ref.[34]), so that the n -cyclecoefficient reduces to: c ( n ) d → s n − πd ( n − (cid:18) nn − (cid:19) n/ (cid:20) n n − ( n − n − (cid:21) d/ , (6)from which we see that closed loops of any number n of nodes are exponentially small as d → ∞ , because thequantity within square brackets is less than unity for n ≥ n -cycle coefficient for the caseof polydisperse spheres. Using Eq. (2) for the connected-ness function, we generalize Eq. (4) as follows: h c d i ( n ) = hC ( n ) i ,...,i n i i ,...,i n hV ( n ) i ,...,i n i i ,...,i n , (7)where C ( n ) i ,...,i n = Z dr ( n ) f i i ( | r − r | ) f i i ( | r − r | ) · · ·· · · × f i n i ( | r n − r | ) , (8) V ( n ) i ,...,i n = Z dr ( n ) f i i ( | r − r | ) f i i ( | r − r | ) · · ·· · · × f i n − i n ( | r n − − r n | ) , (9)and h ( · · · ) i i ,...,i n = X i ,...,i n x i x i · · · x i n ( · · · ) (10)denotes a multiple average over the radii R i , R i , . . . , R i n . In the appendix, we show thatfor bounded distributions of radii, the n -cycle coefficientin the limit d → ∞ is such that: h c d i ( n ) ≤ c ( n ) d χ ( n ) d , (11)where c ( n ) d is the n -cycle coefficient for identical radii,Eq. (6), and χ ( n ) d ∝ d a , where a is a nonnegative constant.Since the exponential decay of c ( n ) for d → ∞ is strongerthan the power-law increase of χ ( n ) d , we see thus thatalso for the case of polydisperse spheres for bounded radiidistributions, the n -cycle coefficient vanishes for any n ≥ IV. SIZE OF FINITE COMPONENTS
The observation made in the previous section thatclosed loops are irrelevant in the large dimensional limitof the Boolean model allows us to consider the compo-nents of the associated weighted random geometric graphas effectively having a tree-like structure. This leads to aconsiderable simplification, as we can take the formalismof the theory of random graphs (see, e.g., Refs. [35–37])and generalize it to the case in which nodes have weights.
A. Multidegree distributions
We start by considering the multidegree distribu-tion of a node of type i , defined as the probability p i (1 , k ; 2 , k ; . . . ) that a sphere of radius R i is connectedto k spheres of radius R , k spheres of radius R ,and so on. Since the radii are randomly and indepen-dently distributed among the N nodes, p i (1 , k ; 2 , k ; . . . ) is just a product of binomial distributions p ij ( k j ) (with j = 1 , , . . . ), each giving the probability that k j spheresof radius R j overlap the sphere of radius R i : p i (1 , k ; 2 , k ; . . . ) = Y j p ij ( k j ) , (12)with p ij ( k j ) = (cid:18) N j − δ i,j k j (cid:19) ( v ij ex ) k j (1 − v ij ex ) N j − δ i,j − k j , (13)where N j (with j = 1 , , . . . ) is the number of spheresof radius R j , v ij ex are the overlap probabilities given inEq. (3), and δ i,j is the Kronecker symbol.We next consider for all i the limit N i → ∞ such that N i /V = x i ρ remains finite, where ρ = N/V is the to-tal number density. In this limit, Eq. (13) reduces to aPoisson distribution: p ij ( k j ) = z k j ij k j ! e − z ij , (14)where z ij = X k kp ij ( k ) = x j ρ Ω d ( R i + R j ) d (15)is the average number of spheres with radius R j thatoverlap a given sphere of radius R i .In addition to the node degree distribution p i (1 , k ; 2 , k ; . . . ), for the following analysis wewill also need the excess node degree distribution q ji (1 , k ; 2 , k ; . . . ), defined as the conditional probabilitythat a sphere of radius R j is connected to k l spheres ofradius R l (with l = 1 , , . . . ), given that it is connectedto a sphere of radius R i . This task is simplified by theirrelevance of closed loops in the large dimensionalitylimit. In this case, indeed, if we select at random anedge connecting a node of type j with a node of type i , the j node attached to the edge is k i times morelikely to have degree k i than degree 1 with nodes of type i . Its degree distribution will thus be proportional to k i p j (1 , k ; 2 , k ; . . . ). The excess degree distribution of a j node that has k i edges with nodes of type i other thanthe edge with the node i to which is attached is thus[38]: q ji (1 , k ; 2 , k , . . . ) = ( k i +1) p j (1 , k ; . . . ; i, k i +1; . . . ) P k ( k i +1) p j (1 , k ; . . . ; i, k i +1; . . . ) , (16)where P k = P k ,k ,... . From Eqs. (12) and (14), q ji (1 , k ; 2 , k , . . . ) reduces simply to: q ji (1 , k ; 2 , k , . . . ) = ( k i + 1) p ji ( k i + 1) z ji Y l = i p jl ( k l )= Y l p jl ( k l ) , (17)where we have used ( k i +1) p ji ( k i +1) = z ji p ji ( k i ). Equa-tion (17) states thus the well-known property that theexcess degree distribution coincides with the node degreedistribution when this is Poissonian [35]. FIG. 2. (Color online) Schematic representation of a finitetree-like cluster formed by connected spheres of radii R , R ,and R . Each node label corresponds to the value of theradius of the sphere attached to the node. B. Mean cluster size in the subcritical regime
We exploit now the statistical irrelevance of closedloops discussed in Sec. III to find the mean size S offinite clusters of connected spheres as d → ∞ . In doingso, we shall first keep the form of the degree distributionsunspecified, and apply Eqs. (14) and (17) only at the endof the calculation.Let us start by considering a randomly selected nodethat has probability x i of being occupied by a sphere ofradius R i . Due to the general tree-like structure of thegraph, the cluster to which the selected node belongs isformed by branches attached to the node according tothe degree distribution p i (1 , k ; 2 , k ; . . . ), as schemati-cally shown in Fig. 2. The mean size S i of the cluster towhich the selected node belongs is thus: S i = x i + x i X k p i (1 , k ; 2 , k ; . . . ) X j k j T ij , (18)where T ij is the mean cluster size of one of the k j branches attached to the selected node. Since the clus-ters have a tree-like structure, T ij is given by the mass(unity) of one neighbor of the selected node, plus themean cluster size of each of the remaining subbranchesattached to the neighbor. To find T ij , we thus need theexcess degree distribution q ji (1 , k ; 2 , k ; . . . ) of a sphereof radius R j connected to the selected node of type i : T ij = 1 + X k q ji (1 , k ; 2 , k ; . . . ) X l k l T jl . (19)Equations (18) and (19) are quite general, as they ap-ply also to tree-like graphs with degree distributions that are not reducible to a multiplication of Poissonian prob-abilitites. Interestingly, similar equations are found inthe calculation of finite size components of multigraphs(also denoted multiplex networks), formed by differentnetworks, each having particular node properties, cou-pled together [38, 39]. The Boolean-Poisson model withrandom radii can thus be viewed also as a particular typeof multigraph, in which each individual network is con-stituted by nodes occupied by spheres of a given radius.Let us now use the results of Sec. IV A and rewriteEqs. (18) and (19) by substituting p i (1 , k ; 2 , k ; . . . ) and q ji (1 , k ; 2 , k ; . . . ) with, respectively, Eqs. (12) and (17): S i = x i + x i X j X k kp ij ( k ) T j = x i + x i X j z ij T j , (20) T j = 1 + X l X k kp jl ( k ) T l = 1 + X l z jl T l , (21)where we have used Eq. (15) and the fact that T ij de-pends only on the neighbor ( j ) of the selected node, i.e., T ij = T j .The mean cluster size is given by S = P i S i , whichfrom Eq. (20) reduces to: S = 1 + P ij x i z ij T j . This rela-tion is obtained also if we multiply both sides of Eq. (21)by x j and sum over j . We can thus write: S = X j x j T j , (22)which states that S is just the average over the sphereradii of the mean cluster size of the branches. C. Equivalence with the Ornstein-Zernike equationfor the pair-connectedness
In continuum percolation theory, cluster statistics areoften studied using the formalism of pair-connectednesscorrelation functions [3, 40, 41], which exploits well devel-oped techniques of liquid state theory. This method hasbeen recently used to studying percolation of monodis-perse spheres in large dimensions [28].As long as closed loops can be neglected, the networkformalism discussed above and the pair-connectednessfunctions method give identical results, provided that thesecond-virial approximation is taken. To see how thisequivalence holds true for the Boolean model in largedimensions, let us first consider the pair-connectednessfunction P ij ( r − r ′ ), defined such that x i x j ρ P ij ( r − r ′ ) d r d r ′ is the probability of finding two spheres of radii R i and R j within the volume elements d r and d r ′ cen-tered respectively in r and r ′ , given that they belong tothe same cluster. The mean cluster size S is given interms of P ij ( r − r ′ ) by the following relation[42]: S = 1 + ρ X i,j x i x j P ij , (23)where P ij = R d r P ij ( r ). P ij is the solution of the pairconnectedness analog of the Ornstein-Zernike equation ofthe liquid state theory of fluids: P ij = C ij + ρ X l x l C il P lj , (24)where C ij = R d r C ij ( r ) is the volume integral of the di-rect pair connectedness function C ij ( r ), which describesshort-range connectivity correlations. Let us introducethe quantity e T i defined as: e T i = 1 + ρ X j x j P ij . (25)The use of the above expression reduces Eq. (23) to: S = X i x i + ρ X i,j x i x j P ij = X i x i ρ X j x j P ij = X i x i e T i , (26)while inserting Eq.(24) into Eq. (25) leads to: e T i = 1 + ρ X j x j C ij + ρ X l x l C il P lj ! = 1 + ρ X j x j C ij + ρ X j x j C ij ( e T j − ρ X j x j C ij e T j . (27)We see that Eqs. (26) and (27) are identical to respec-tively Eqs. (22) and (21) if we identify ρx j C ij with z ij .From Eq. (15), we obtain thus: C ij = z ij ρx j = Ω d ( R i + R j ) d , (28)which corresponds to take the volume integral ofthe second-virial approximation C ij ( r ) = C (2) ij ( r ) = f ij ( r ) for the direct pair-connectedness function. Thisis not surprising, because in the density expansionof the direct pair-connectedness function, C ij ( r ) = P n ≥ ρ n − C ( n ) ij ( r ), the terms with n ≥ V. UNIVERSALITY OF THE PERCOLATIONTHRESHOLD
We proceed to find the percolation threshold for theBoolean-Poisson model of polydisperse spheres in thelarge dimensionality limit. We shall consider the caseof bounded distributions of the radii, for which we haveshown in Sec. III that closed loops of connected particlescan be neglected for d → ∞ , and Eqs. (21) and (22) are valid. To measure the sphere concentration we introducethe dimensionless density η = ρ Ω d h R d i R = ρ Ω d X i x i R di . (29)The percolation threshold η c is defined as the smallestvalue of η such that S diverges. This definition is equiv-alent to finding the smallest pole of Eq. (21), if it exists. A. Discrete radii distributions
We first consider the case in which the spheres have afinite number M of radii: ρ ( R ) = M X i =1 x i δ ( R − R i ) , (30)so that using Eqs. (15), (21) and (22) we rewrite theequations for the mean cluster size as: S = M X i =1 x i T i , (31) T i = 1 + ρ Ω d M X j =1 x j ( R i + R j ) d T j . (32)Without loss of generality, we assume that R M is strictlythe largest radius out of the M possible values of theradii, and we introduce q i = R i /R M , which takes valuessmaller than the unity for all i = M . For large d , thedimensionless density η reduces to: η = ρ Ω d M X i =1 x i R di = ρ Ω d R dM " x M + M − X i =1 x i q di → ρ Ω d R dM x M , (33)because q di goes exponentially to zero as d → ∞ when i = M , and Eq. (32) becomes: T i = 1 + 2 d η x M X j x j (cid:18) q i + q j (cid:19) d T j . (34)We note that (cid:16) q i + q j (cid:17) d is vanishingly small as d → ∞ unless i = j = M , for which it takes the value 1. Thesmallest pole of Eq. (34) for large d is thus the solutionof: T i = 1 + 2 d ηT M δ i,M , (35)where δ i,j is the Kronecker delta. Equation (35) is solvedby T M = 1 / (1 − d η ) and T i = 1 for i = M , so that themean cluster size (31) becomes: S = M − X i =1 x i + x M T M = x M − d η , (36) d η c z c (a) (b) FIG. 3. (Color online) (a) Percolation threshold η c in units ofthe asymptotic value 2 − d as a function of dimensionality fora discrete distribution of radii with M = 2 and R /R = 1 / η c can be calculated exactly for any d . x = 0 .
8, 0.6 , . . z c .As d → ∞ , z c tends asymptotically to x . which diverges when η → η c = 12 d . (37)The above expression for η c holds true for any sequenceof occupation fractions x i , independent of dimensional-ity, provided that x M = 0. In particular, Eq. (37) con-firms and extends to M > η c = 1 / d is also the limit for infinite dimensions ofthe percolation threshold of monodisperse spheres withradius R M , whose mean cluster size is given by Eq. (36)with x M = 1.The origin of the universality of η c can be traced backto the divergence of T M , which indicates that the onset ofa giant component of connected polydisperse spheres isestablished only by the subset of spheres with the maxi-mum radius when d → ∞ . In other words, at d → ∞ thecontribution to percolation of the smaller spheres van-ishes, and the resulting η c is the critical threshold for asystem of monodisperse spheres of radius R M . Follow-ing the observation that systems of polydisperse sphereswith M different radii can be interpreted as a multinet-work of coupled M subnetworks (see Sec. IV B), we seethat Eq. (35) is equivalent to decoupling the subnetworksassociated with each radius, and that long-range connec-tivity arises only from the network formed by sphereswith radius R M .One interesting consequence of the irrelevance ofsmaller radii at percolation is that the critical averageconnectivity per particle, z c = X i,j x i x j ρ c Ω d ( R i + R j ) d , (38) reduces for d → ∞ to: z c = 2 d ρ c Ω d R dM X i,j x i x j (cid:18) q i + g j (cid:19) d → d η c x M = x M , (39)where ρ c is the critical number density. For x M <
1, thecritical average connectivity is thus less than the unityfor d → ∞ , which must be contrasted to z c ≥ M = 2), Eq. (32) reduces to asystem of two linear equations that can be solved exactlyfor any d . The resulting η c and z c are shown in Figs. 3(a)and 3(b), respectively, for R = 2 R and different valuesof the fraction x of spheres of radius R . The asymptoticlimits η c = 2 − d and z c = x are recovered for sufficientlylarge values of d . B. Continuous radii distributions
Let us now consider the case in which the radii distri-bution ρ ( R ) is a continuous bounded function indepen-dent of d . We again denote by R M < ∞ the maximumallowed radius, so that ρ ( R ) = 0 for R > R M , and werewrite the equations for the mean cluster size in termsof continuous variables of the radii: S = h T ( R ) i R , (40) T ( R ) = 1 + ρ Ω d h ( R + R ′ ) d T ( R ′ ) i R ′ , (41)where h ( · · · ) i R = R R M dRρ ( R )( · · · ). We expand the bi-nomial power ( R + R ′ ) d and use η = ρ Ω d h R d i R to write: T ( R ) = 1 + η d X k =0 (cid:18) dk (cid:19) R d − k h R d i R h R k T ( R ) i R . (42)If we multiply both sides of Eq. (42) by R n / h R n i R , with n = 1, 2, . . . , d , and average over R we arrive at: t ( n ) = 1 + η d X k =0 (cid:18) dk (cid:19) h R n + d − k i R h R k i R h R d i R h R n i R t ( k ) , (43)where t ( n ) = h R n T ( R ) i R h R n i R . (44)From Eqs. (40) and (44) we see that the mean clustersize can be obtained from S = t (0).To solve Eq. (43), we note that for large d the binomialcoefficient is strongly peaked at k = d/ (cid:18) dk (cid:19) ≃ d r πd e − d ( k − d/ = 2 d +1 d g (cid:18) kd − , √ d (cid:19) , (45)where g ( x, σ ) = exp( − x / σ ) / √ πσ is the Gaussianfunction. Provided that the radii distribution is bounded,the binomial coefficent dominates the k -dependence ofthe kernel. To see this, let us consider the m -th moment h R m i R = R mM R dyρ ( y ) y m , where y = R/R M . For large m the main contribution to the integral comes from y close to 1. Thus we make the quite general assumptionthat for y →
1, the radii distribution behaves as ρ ( y ) ∝ (1 − y ) α − , with α >
0. Setting t = m (1 − y ) for large m , we find h R m i R ∝ R mM m α Z m dt t α − (1 − t/m ) m ≃ R mM m α Z ∞ dt t α − e − t = R mM m α Γ( α ) , (46)so that for large k the term h R n + d − k i R h R k i R in Eq. (43)is proportional to R n + dM / [( n + d − k ) k ] α , which has a muchweaker k -dependence than Eq. (45). Next, we introduce s = 2 n/d and s ′ = 2 k/d , which we treat as continuousvariables for d → ∞ , and we replace the sum over k by an integral over s ′ : P dk =0 → d R ds ′ . If we denote˜ t ( s ) = t ( ds/
2) and ˜ t ( s ′ ) = t ( ds ′ / t ( s ) = 1 + 2 d η Z ds ′ (cid:20) g (cid:18) s ′ − , √ d (cid:19) × h R d [1+( s − s ′ ) / i R h R ds ′ / i R h R d i R h R ds/ i R ˜ t ( s ′ ) . (47)Since g ( s ′ − , / √ d ) → δ ( s ′ −
1) for d → ∞ , the aboveexpression reduces to:˜ t ( s ) = 1 + η d h R d (1+ s ) / i R h R d/ i R h R d i R h R ds/ i R ˜ t (1) , (48)from which we obtain the mean cluster size: S = ˜ t (0) = 1 + η d h R d/ i R h R d i R ˜ t (1) . (49)Setting s = 1 in Eq. (48), we find ˜ t (1) = (1 − d η ) − , sothat we arrive finally at: S = 11 − d η h R d/ i R h R d i R , (50)which, as found for the case of discrete distributions, di-verges at η → η c = 12 d , (51)independently of the particular form of the bounded dis-tribution ρ ( R ).Using Eq. (45) and considering the weak dependenceof the moments of R , we readily obtain the large dimen- d η c
10 100d10 -2 -1 z c (a) (b) FIG. 4. (Color online) (a) Dimensional dependence of thepercolation threshold η c in units of the asymptotic value 2 − d obtained from a numerical solution of Eq. (43) for rectangu-lar (upper curve), semicircular (middle curve), and triangular(lower curve) distributions of the sphere radii. (b) Corre-sponding critical average connectivity per particle z c (solidcurves). Dashed lines are the asymptotic results for d ≫ z c = 4 /d (rectangular distribution), z c = 32 / ( √ πd / ) (semi-circular distribution), and z c = 32 /d (triangular distribu-tion). sional limit of the critical average connectivity per parti-cle: z c = ρ c Ω d h ( R + R ′ ) d i R,R ′ = η c d X k =0 (cid:18) dk (cid:19) h R k i R h R d − k i R h R d i R → h R d/ i R h R d i R , (52)from which we see that z c ≤ z c = x M when ρ ( R ) is given by Eq. (30).We complete this section by showing how the per-colation threshold obtained from Eqs. (40) and (41)evolves towards the asymptotic value η c = 2 − d as d increases. Toward that end, we consider radii dis-tributions of rectangular, semicircular, and triangularshapes, given respectively by ρ ( R ) = 1 /R M , ρ ( R ) =4 p ( R M / + ( R − R M / /π , and ρ ( R ) = 2( R M − R ) /R M , for R ≤ R M and zero otherwise. We calculate η c from the smallest pole of S = t (0) obtained from a nu-merical solution of Eq. (43). The resulting thresholds arevery close to 2 − d for all d considered, and they approachthe asymptotic limit from below, as shown in Fig. 4(a).For the same cases of Fig. 4(a), we have calculated alsothe d -dependence of z c , shown in Fig. 4(b) by solid lines,which we compare with the asymptotic limits (dashedlines) z c = 4 /d , z c = 32 / ( √ πd / ), and z c = 32 /d ob-tained from Eq. (52) for rectangular, semicircular, andtriangular distributions of the radii, respectively. VI. THE CASE OF UNBOUNDEDDISTRIBUTION OF THE RADII
Having established that η c is universal as d → ∞ forbounded (and independent of d ) distributions of the radii,it is natural to ask if universality holds true also when ρ ( R ) is unbounded. Although we have shown the ir-relevance of closed loops limited to the case of bondeddistributions, we shall nevertheless assume that n -cyclecoefficients are negligible also for unbounded ρ ( R ), andthat graph components have a tree-like structure. Letus consider the specific case of a lognormal distributionfunction: ρ ( R ) = 1 √ πσR exp (cid:20) − ln ( R/R )2 σ (cid:21) , (53)where R ∈ [0 , ∞ ), R is the median radius, and σ is thestandard deviation of ln( R ). Equation (53) is an interest-ing case-study, as the resulting η c and z c for asymptot-ically large d can be found analytically. Using the k -thmoment h R k i R = R k exp( σ k / t ( n ) = 1 + η d X k =0 (cid:18) dk (cid:19) e σ [( n + d − k ) + k − n − d ] t ( k ) , (54)from which we express the mean cluster size as: S = t (0) = 1 + η d X k =0 (cid:18) dk (cid:19) e σ k ( k − d ) t ( k ) . (55)For sufficiently large d , the only nonvanishing terms ofthe summation are those with k = 0 and k = d , so that: S = 1 + η [ S + t ( d )] , (56)where from Eq. (54) t ( d ) is given by: t ( d ) = 1 + η d X k =0 (cid:18) dk (cid:19) e σ ( d − k ) t ( k ) . (57)For d → ∞ , t ( d ) tends asymptotically to t ( d ) = 1 + ηe σ d t (0), as the term with k = 0 dominates the sumover k in Eq. (57). We thus find that the mean clustersize, Eq. (56), reduces to: S = 1 + η − η − η e σ d , (58)which diverges at the asymptotical critical value, η → η c = e − σ d . (59)The corresponding critical coordination number is z c = η c d X k =0 (cid:18) dk (cid:19) h R k i R h R d − k i R h R d i R = η c d X k =0 (cid:18) dk (cid:19) e σ k ( k − d ) → η c , (60) -2 -1 e x p ( σ d / ) η c z c / η c (a) (b) FIG. 5. (Color online) (a) Dimensional dependence of thepercolation threshold η c in units of the asymptotic valueexp( − σ d /
2) for a lognormal distribution of the radii ob-tained from a numerical solution of Eq. (54); σ = 0 .
25, 0 . .
4, and 0 . z c for σ = 0 .
25, 0 . .
4, and 0 . z c /η c → d → ∞ . where we have again used the fact that for large d onlythe terms k = 0 and k = d contribute to the summation.As evidenced in Eq. (59), the percolation thresholdfor infinite dimensions is no longer universal, as it de-pends on the parameter σ of the log-normal distribu-tion. Interestingly, from Eq. (59) we also see that η c canbe smaller than the critical threshold of monodispersespheres ( η c = 2 − d ), contrary to what is expected in finitedimensions [24]. We note that a critical threshold smallerthan the monodisperse sphere limit in large dimensionshas been found also for the case of radii distributionswith d -dependent weights [25, 26].To verify the accuracy of Eq. (59), we compare it withthe threshold obtained by solving numerically Eq. (54).As d increases, the asymptotic limit η c = e − σ d isreached more rapidly when σ is larger, as shown inFig. 4(a). From inspection of Eq. (55) we see that thisbehavior is due to the competition between e σ k ( k − d ) andthe maximum value ∼ d of the binomial coefficient at k = d/
2: the latter is suppressed by the exponential func-tion when d > /σ . From numerical calculation of z c , shown in Fig. 5(b) for the same σ values of Fig. 5(a),we see that also the asymptotic formula for z c , Eq. (60),is verified. VII. LOWER BOUND ON THE PERCOLATIONTHRESHOLD
Having established that Eqs. (21) and (22) give asymp-totic limits of the critical threshold η c as d → ∞ , weshow now that the same equations provide also a lowerbound on η c for any dimensionality. Toward that end, wetake the pair-connectedness function P ij ( r ) considered inSec. IV C, and we extend to the polydisperse sphere casethe inequality formulated in Ref. [43]: P ij ( r ) ≤ f ij ( r ) + ρ X l x l Z d r ′ f il ( | r − r ′ | ) P lj ( r ′ ) , (61)where f ij ( r ) is the connectedness function given inEq. (2). The above expression applies to any dimension-ality, and following Ref. [28], where Eq. (61) has beenused for the monodisperse sphere case, it enable us tofind a lower bound on the percolation threshold. To seethis, we take the volume integral of Eq. (61), P ij ≤ V ij ex + ρ X l x l V il ex P lj , (62)where V ij ex = R d r f ij ( r ) = Ω d ( R i + R j ) d , and we useEqs. (25) to find: e T i ≤ ρ X j x j V ij ex e T j = 1 + X j z ij e T j , (63)which together with Eq. (26) gives an upper bound forthe mean cluster size: S = X i x i e T i ≤ X i x i T i , (64)where T i is the solution of Eq. (21). From the inequal-ity of Eq. (64), we see that the value of η such that P i x i T i diverges identifies a lower bound on the perco-lation threshold for any d . The solid lines plotted inFigs. 3(a)-5(a) represent thus lower bounds on η c for thedifferent radii distribution functions considered in thiswork. As d increases, these lower bounds tend asymptot-ically to the infinite dimensional limit 2 − d for boundedradii distributions and to exp( − σ d /
2) for lognormalradii distributions. Finally, we note that Eq. (64) im-plies also that the values of z c shown in Figs. 3(b)-5(b)are lower bounds on the critical connectivity for any di-mensionality. VIII. SUMMARY AND DISCUSSION
We have considered random dispersions of penetrable d -dimensional spheres with distributed radii in terms ofweighted random geometric graphs, where nodes repre-sent sphere centers and edges connect nodes of overlap-ping spheres with probability weighted by the sphereradii. For bounded distribution of the radii, we haveshown that closed loops of connected spheres can be ne-glected in the limit d → ∞ and that graph componentshave thus tree-like structure. Analysis of the mean clus-ter size reveals that the asymptotic percolation thresholdis universal and coincides with the threshold η c = 2 − d found for the case of monodisperse spheres in high di-mensions. This result confirms and extends a previousfinding on the percolation of d → ∞ spheres with twodifferent radii [25, 26]. Furthermore, we show that the asymptotic critical connectivity per particle z c , thoughdependent on the shape of the radii distribution function,is less than unity and approaches z c → d -independent log-normal function, which is a treatable example of un-bounded distribution. Assuming that clusters have atree-like structure, we find that the percolation threshold η c depends on the shape of the log-normal distributionand, interestingly, that η c for d → ∞ can be smaller thatthe threshold for monodisperse spheres, in contrast towhat is expected at finite dimensions [24].Before concluding, let us speculate on the percolationthreshold in homogeneous fluids of polydisperse sphereswith impenetrable cores (cherry-pit model [3]). In finitedimensions, correlations between the cores preclude writ-ing the multi-degree distribution as a product of Pois-son distributions, as done in Sec. IV A, because the N -particle distribution function g N ( r , r , . . . , r N ) dependson the relative positions of the core centers [32]. However,in the limit of infinite dimensions and for small densities, g N ( r , r , . . . , r N ) asymptotically factorizes into a prod-uct of θ -functions that are unity for pair distances beyondthe hard-core diameter [44]. The multi-degree distribu-tion for d → ∞ can thus still be written as a product ofPoisson distributions, with the average number of con-tacts unaltered by the presence of the hard-cores if thepenetrable shells are non-vanishing. With the same rea-soning, closed loops are expected to be negligible andgraphs are still dominated by tree-like components. Fornon-zero penetrable shells, therefore we expect the sameasymptotic results for η c as those obtained for the caseof penetrable hyperspheres. ACKNOWLEDGMENTS
I am grateful to Avik P. Chatterjee, J.-B. Gou´er´e, andS. Torquato for useful comments and suggestions.
Appendix A: Irrelevance of h c d i ( n ) for d → ∞ In this appendix, we show that when the radii distri-bution is independent of d and bounded [that is, when ρ ( R ) = 0 for any R > R M , with R M < ∞ ], the n -cyclecoefficient for polydisperse spheres, defined in Eqs. (7)-(9), vanishes for d → ∞ .Since R M is the maximum radius of the distribution,the connectedness functions in the integrand of Eq. (8)are such that f ij ( r ) ≤ f ( r ) = θ (2 R M − r ) for any i and j . We can thus write: C ( n ) i ,...,i n ≤ Z dr ( n ) f ( | r − r | ) f ( | r − r | ) · · · f ( | r n − r | ) , (A1)0which, when substituted in Eq. (7), gives: h c d i ( n ) ≤ c ( n ) d V V n − hV ( n ) i ,...,i n i i ,...,i n , (A2)where V ex = 2 d Ω d R dM , and c ( n ) d is the n -cycle coefficientfor identical radii given in Eq. (6). Next, we perform theintegrations over r , . . . , r n in Eq. (9) to find: V ( n ) i ,...,i n = V n − Y j =1 V i j ,i j +1 ex = V Ω n − d n − Y j =1 ( R i j + R i j +1 ) d = V Ω n − d n − Y j =1 d X k j =0 (cid:18) dk j (cid:19) R k j i j R d − k j i j +1 , (A3)where in the last equality we have expanded the binomialpowers. In performing the average over R i , . . . , R i n , wemust group the contributions with equal radius variablesand average them independently of the other radii. De- noting a general m -th moment as h R m i R , we obtain: hV ( n ) i ,...,i n i i ,...,i n = V Ω n − d d X k =0 (cid:18) dk (cid:19) · · · d X k n − =0 (cid:18) dk n − (cid:19) × h R k i R h R d − k + k i R h R d − k + k i R · · ·· · · h R d − k n − + k n − i R h R d − k n − i R . (A4)Following Sec. V B, we approximate for large d the bino-mial coefficients by Gaussian functions centered at d/ d → ∞ , P dk j =0 (cid:0) dk j (cid:1) → d R ds j δ ( s j − s j = 2 k j /d and j = 1 , . . . , n −
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