Newman-Ziff algorithm for the bootstrap percolation: application to the Archimedean lattices
NNewman-Ziff algorithm for the bootstrap percolation: application to theArchimedean lattices
Jeong-Ok Choi a , Unjong Yu b, ∗ a Division of Liberal Arts and Sciences, Gwangju Institute of Science and Technology, Gwangju 61005, South Korea b Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju 61005, South Korea
Abstract
We propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending theNewman-Ziff algorithm of the classical percolation [M. E. J. Newman and R. M. Ziff, Phys. Rev. Lett. 85 (2000) 4104].Using these algorithms and the finite-size-scaling, we calculated with high precision the percolation threshold and criticalexponents in the eleven two-dimensional Archimedean lattices. We present the condition for the continuous percolationtransition in the bootstrap percolation and the diffusion percolation, and show that they have the same critical exponentsas the classical percolation within error bars in two dimensions. We conclude that the bootstrap percolation and thediffusion percolation almost certainly belong to the same universality class as the classical percolation.
Keywords:
Percolation, Bootstrap percolation, Newman-Ziff algorithm, Critical exponent, Universality class,Archimedean lattice
1. Introduction
The bootstrap percolation (BP) model [1] has attractedcontinuous attention for its various applications such asdisordered dilute magnetic systems [1, 2], neuronal activity[3], jamming transition [4], and diffusion of innovations[5]. The BP process operates as follows: (i) Each site isoccupied with the probability p and empty otherwise; (ii)Occupied sites that have less than m occupied neighborsbecome empty, and the process is repeated until all theoccupied sites have at least m occupied neighbors. Thediffusion percolation (DP) model [6] is closely-related tothe BP, and sometimes it is also called the BP [7, 8, 9].In the DP, process (i) is the same as the BP, but process(ii) is different: empty sites that have at least k occupiedneighbors become occupied recursively, until all the emptysites have less than k occupied neighbors.The BP and DP have been studied on lattices [10, 11,12, 13, 6, 14, 15, 16, 17, 18, 8], trees [1, 19, 20], and com-plex networks [21, 7, 22, 9]. A few facts are known aboutthe BP and DP. Clearly, the BP with m = 0 and DP with k > ∆ max are the same as the classical percolation (CP)model, where ∆ max is the maximum value of degree. Fora given graph, the percolation threshold of the DP ( p k DP )is not larger than that of the CP ( p CP ), and that of the BP( p m BP ) is not less than that of the CP. The BP with m = 1or m = 2 has the same percolation threshold as the CP. Asfor ∆-regular lattices, if m + k = ∆+1, then there is a closerelationship between the BP and DP: (1) The sum of per-colation thresholds of the BP and its corresponding DP is ∗ Corresponding author: Tel.: +82-62-715-3629;
Email address: [email protected] (Unjong Yu) the same or larger than 1 ( p m BP + p ∆+1 − m DP ≥
1) [6] (The sumis 1 for self-matching lattices such as the triangular lattice[23].); (2) There are lattice-dependent parameters m c and k c , where the BP of m and DP of k have first-order perco-lation transitions with the percolation thresholds p m BP = 1and p k DP = 0, respectively, if and only if m > m c and k < k c ; the two parameters satisfy m c + k c = ∆ + 1. Forthe square and triangular lattices, m c = ∆ / m = 1 isknown to have the same critical exponents as the CP [14],it is not clear for the DP and the BP of m ≥
2. Kogut andLeath argued that β depends on m for the BP from Monte-Carlo simulations on the square, triangular, and cubic lat-tices [10], and renormalization group studies confirmed it[11, 13]. Adler conjectured that ν is universal but β is non-universal in the BP and DP [6, 14, 15]. To the contrary,other studies, which include most recent simulations, insistthat both ν and β are universal [12, 16, 17, 18]. They ar-gued that non-universality of previous works can be fromthe small size of clusters used in the simulations. How-ever, we judge that the universality class of the BP andDP is not definitely clear yet. In Ref. [17], for example, theFisher exponent τ for the DP with k = 4 was calculated onthe triangular lattice to be τ = 2 . ± .
04, which is con-sistent with τ = 187 / ≈ .
055 for the two-dimensionalCP. At first glance, it looks like a good evidence to sup-
Preprint submitted to Elsevier March 6, 2019 a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r ort the conclusion of the same universality; however, β obtained by the scaling relation β/ν = 2( τ − / ( τ −
1) intwo dimensions is β/ν = 0 . ± .
08, which is consistentwith β/ν = 5 / ≈ .
104 of the CP but the uncertainty istoo large to make any conclusion. In addition, the BP andDP have been studied only in four kinds of lattices (thesquare, triangular, honeycomb, and cubic lattices).In this paper, we introduce efficient algorithms for theBP and DP models and present much more precise re-sults on eleven two-dimensional Archimedean lattices. Wecalculated percolation threshold and critical exponents ( ν and β ) to present positive evidences for that the BP andDP belong to the same universality class as the CP in twodimensions.
2. Methods
The classical site percolation model can be simulatedsimply by the following two steps: (i) Fill each site withindependent probability p and leave it empty with prob-ability 1 − p ; (ii) Identify all the connected clusters tocheck whether a percolating cluster exists. The simulationshould be repeated many times to make a statistical aver-age for a given p . This algorithm is simple but inefficient.In order to get results with different probability p (cid:48) , thewhole simulation should be done again. It is more difficultto calculate the derivative of a quantity with respect to theprobability p , because numerical differentiation inevitablygives large error [26]. A more efficient algorithm was pro-posed by Newman and Ziff [27, 28], and it has becomea standard method in classical percolation studies. TheNewman-Ziff algorithm consists of four steps: (i) Initially,all sites are empty; (ii) Choose an empty site randomlyand fill it; (iii) Update the information of connected clus-ters to check whether percolation occurs; (iv) Repeat steps(ii) and (iii) until all the sites are occupied. An efficientalgorithm to update the information of connected clus-ters (tree-based union/find algorithms) is also presentedin Ref. [28]. An average (cid:104) Q ( n ) (cid:105) is obtained by repeatingthe whole steps, where Q ( n ) is any quantity (e.g., size ofthe largest cluster) for a fixed number of occupied sites n .In one run of the Newman-Ziff algorithm, Q ( n ) is corre-lated with Q ( n ) inevitably, but values of Q ( n ) of differ-ent runs are absolutely independent and so the statisticalaveraging of (cid:104) Q ( n ) (cid:105) has no problem. A value (cid:104) Q ( p ) (cid:105) fora fixed occupation probability p can be obtained by thetransformation of (cid:104) Q ( p ) (cid:105) = N (cid:88) n =0 N ! n !( N − n )! p n (1 − p ) N − n (cid:104) Q ( n ) (cid:105) , (1)where N is the total number of sites. Therefore, once (cid:104) Q ( n ) (cid:105) is obtained, (cid:104) Q ( p ) (cid:105) can be calculated for all valuesof p . Another advantage of the Newman-Ziff algorithm isthat the derivative can be obtained through d (cid:104) Q ( p ) (cid:105) dp = ddp (cid:34) N (cid:88) n =0 N ! n !( N − n )! p n (1 − p ) N − n (cid:104) Q ( n ) (cid:105) (cid:35) (2) = N (cid:88) n =0 N ! n !( N − n )! p n − (1 − p ) N − n − ( n − N p ) (cid:104) Q ( n ) (cid:105) (3)without numerical differentiation [29].However, the Newman-Ziff algorithm cannot be useddirectly in the BP or DP models, because filling of eachsite depends on the local environment. As for the DP, theNewman-Ziff algorithm can be modified as follows.(1) Initially, all sites are empty.(2) Make an array of all the sites in random order.(3) Get one site by the array. If the site is empty, fill itand other sites that have at least k occupied neighbors,recursively. If the site is already occupied, do nothing.(4) Update the information about connected clusters tocheck whether percolation occurs.(5) Repeat steps (3) and (4) until all the sites are occupied.The whole steps are repeated to make an average quan-tity (cid:104) Q ( n ) (cid:105) , and the transformation of Eq. (1) gives (cid:104) Q ( p ) (cid:105) .This algorithm is equivalent to the original DP with fixed p because the final state does not depend on the sequence ofthe filling process once an initial occupation is determined.The BP model can be simulated by the same way as DP:Initially all sites are filled and sites are emptied one-by-onein random order. This algorithm is called the avalanch-ing bootstrap percolation of the second kind (ABP2) [30].However, the algorithm is inefficient because it is difficultto identify and update the cluster information during sim-ulation. Therefore, we propose a more efficient algorithmfor the BP model by introducing preoccupied state in addi-tion to empty and occupied states. When a site is chosenin the Newman-Ziff algorithm, it is occupied only whenthere are at least m occupied neighbors; otherwise, it isassigned to be preoccupied. Preoccupied state means thatthe site will be occupied after the condition is satisfied.The algorithm can be presented as follows.(1) Initially, all sites are empty.(2) Make an array of all the sites in random order.(3-a) Get one site by the array. Set the site into the pre-occupied state.(3-b) Identify the connected cluster of preoccupied sitesthat includes the site. (To accelerate the simulation, pre-occupied sites with less than m occupied or preoccupiedneighbors can be excluded from the cluster, because theyare impossible to be filled.)(3-c) Fill tentatively all the sites of the cluster.(3-d) Within the cluster, set sites that have less than m occupied neighbors into the preoccupied state again, recur-sively, until all the occupied sites have at least m occupiedneighbors.(4) Update the information about connected clusters tocheck whether percolation occurs.(5) Repeat steps (3) and (4) until all the sites are occupied.All steps of this algorithm are identical with the Newman-2 ) T2 (4 ) T3 (6 ) T4 (3 ,6) T5 (3 ,4 ) T6 (3 ,4,3,4)T7 (3,4,6,4) T8 (3,6,3,6) T9 (3,12 ) T10 (4,6,12) T11 (4,8 ) Figure 1: The eleven Archimedean lattices. The numbers in parentheses represent the sequence of regular polygons around each vertex.
Ziff algorithm of the CP except for step (3), and the ef-ficient routine that updates the information of connectedclusters can be used without modification. Although wefocus on two-dimensional lattices in this paper, both of thealgorithms for the DP and BP can also be applied to anydimensional systems and to complex networks.The CPU time needed in these algorithms is propor-tional to the lattice size by T cpu ∼ N λ with λ ≈ .
2. CPUtime for the DP, CP, and BP ( m <
3) are of the same orderof magnitude for the same value of N ; in the case of theBP with m = 3, CPU time requirement is about 10 timesmore than that of the CP. Three-dimensional lattices showthe same behavior, but more calculations would be neededthan two dimensions because the number of sites increasesas N ∼ L , where L is linear size. For a lattice of N = 10 ,it takes about one second of CPU time for one sweep ex-cept for the BP of m = 3 by Intel(R) Xeon(R) CPU of2.2 GHz. Note that the running time can be easily re-duced by parallelizing the algorithm. In the case of thetraditional brute force approach, which calculates (cid:104) Q ( p ) (cid:105) directly, the CPU time requirement depends on the num-ber of occupied sites. On average, it takes about half ofthe Newman-Ziff algorithm because some part of the lat-tice is not filled at all. However, it gives only (cid:104) Q ( p ) (cid:105) at aspecific value of p , and another independent calculation isneeded to get (cid:104) Q ( p (cid:48) ) (cid:105) for p (cid:48) (cid:54) = p . In addition, it is practi-cally impossible to get its derivative d (cid:104) Q ( p ) (cid:105) /dp with highprecision. Therefore, our new algorithms are much moreefficient except when only (cid:104) Q ( p ) (cid:105) is to be calculated at aknown specific p .Using these algorithms, we calculated the strength ofthe largest cluster (the probability that a site belongs tothe largest cluster; P ∞ ), average cluster size excluding thelargest one ( M (cid:48) ), percolation probability in any direction( P w ) and in both directions ( P w ), and proportion of oc-cupied sites ( P o ) as a function of p for the CP, BP, andDP.In this work, we consider eleven Archimedean lattices,which are vertex-transitive graphs made in two dimensionsby edge-to-edge tiling of regular polygons whose vertices are surrounded by the same sequence of polygons. Thereare only eleven Archimedean lattices [31] and they are typ-ically used for systematic studies [32, 33, 34]. They areshown in Fig. 1. The periodic boundary condition is usedand the percolation is defined by the existence of a clusterthat wraps all the way around the lattice. The numberof lattice sites studied in this work is from N = 1296 to N = 2 560 000.The percolation thresholds ( p k DP and p m BP ) of infinitelattices are determined by the finite-size-scaling. The per-colation threshold estimate of a finite lattice is determinedby the probabilities of initial filling ( p ) that give the max-imums of physical quantities that show critical behavioror their derivatives. The percolation threshold can also befound by the crossing points of percolation probabilitiesof different lattice size [28]. We averaged the percolationthreshold values obtained by these methods to get the finalestimate. Critical exponents ν and β are obtained by thederivative of percolation probability and P ∞ , respectively[35, 29]. The correction-to-scaling [36, 37] is ignored.Most of the results were produced by using Mersennetwister pseudo-random-number generator (MT19937) [38],which was confirmed reliable in a site percolation problem[39]. We also confirmed that other pseudo-random-numbergenerators give equivalent results.
3. Results
Figure 2 shows the strength of the largest cluster P ∞ ( p, L ),average cluster size excluding the largest one M (cid:48) ( p, L ),percolation probability in any direction P w ( p, L ) and inboth directions P w ( p, L ), and proportion of occupied sites P o ( p, L ) for the DP ( k = 4), CP, and BP ( m = 3) in the tri-angular lattice. Parameter L is the linear size of the lattice.They all show continuous phase transition, which becomessharper as the lattice size increases. The proportion of oc-cupied sites P o ( p, L ) is independent of lattice size and doesnot show any critical behavior at the percolation thresh-old. Equivalent behavior is observed in the BP and DP ofthe other values of m ≤ m c and k ≥ k c , and in the other3 able 1: Name, coordination number (∆), percolation threshold ( p m BP and p k DP ), and critical exponents ( ν and β ) for the BP and DP withthe continuous percolation transition of the eleven Archimedean lattices. Bootstrap percolation Diffusion percolationLattice ∆ m p m BP ν β/ν k p k DP ν β/ν T1 6 1 0.49997(4) 1.336(3) 0.104(1) 6 0.50000(1) 1.335(1) 0.104(1)(3 ) 2 0.49999(4) 1.336(3) 0.104(1) 5 0.49999(2) 1.334(1) 0.104(1)3 0.62915(5) 1.335(3) 0.104(2) 4 0.37083(4) 1.334(2) 0.103(1)T2 4 1 0.59272(3) 1.335(4) 0.104(1) 4 0.54731(1) 1.333(4) 0.104(1)(4 ) 2 0.59272(3) 1.336(3) 0.104(1) 3 0.42037(2) 1.333(2) 0.104(1)T3 3 1 0.69710(5) 1.337(3) 0.104(1) 3 0.56008(1) 1.333(1) 0.104(1)(6 ) 2 0.69703(4) 1.335(3) 0.104(1) 2 0.30943(2) 1.333(2) 0.104(1)T4 5 1 0.57948(3) 1.337(3) 0.104(1) 5 0.57950(1) 1.332(2) 0.104(1)(3 ,
6) 2 0.57948(3) 1.336(3) 0.104(1) 4 0.48450(1) 1.334(1) 0.104(1)3 0.73227(3) 1.333(2) 0.103(3) 3 0.26936(3) 1.334(1) 0.104(1)T5 5 1 0.55020(3) 1.337(3) 0.104(1) 5 0.54387(1) 1.333(1) 0.104(1)(3 , ) 2 0.55020(4) 1.335(3) 0.104(1) 4 0.47648(2) 1.333(2) 0.104(1)3 0.71884(4) 1.330(4) 0.103(1) 3 0.28165(1) 1.332(1) 0.103(1)T6 5 1 0.55080(3) 1.337(4) 0.104(1) 5 0.54108(1) 1.333(1) 0.104(1)(3 , , ,
4) 2 0.55080(4) 1.336(3) 0.104(1) 4 0.47072(1) 1.334(1) 0.104(1)3 0.72813(5) 1.336(4) 0.101(4) 3 0.27194(1) 1.335(1) 0.103(1)T7 4 1 0.62180(3) 1.337(3) 0.104(1) 4 0.57502(1) 1.332(2) 0.104(1)(3 , , ,
4) 2 0.62178(4) 1.336(3) 0.104(1) 3 0.42652(1) 1.333(1) 0.104(1)3 0.86713(5) 1.337(4) 0.103(3) 2 0.13447(3) 1.334(2) 0.104(1)T8 4 1 0.65268(3) 1.336(5) 0.104(1) 4 0.58661(1) 1.334(2) 0.104(1)(3 , , ,
6) 2 0.65269(4) 1.337(4) 0.104(1) 3 0.39451(2) 1.335(2) 0.104(1)T9 3 1 0.80787(3) 1.338(4) 0.104(1) 3 0.65335(1) 1.332(1) 0.104(1)(3 , ) 2 0.80787(3) 1.338(3) 0.104(1) 2 0.34028(4) 1.333(2) 0.104(1)T10 3 1 0.74779(3) 1.339(4) 0.104(1) 3 0.61644(1) 1.332(1) 0.104(1)(4 , ,
12) 2 0.74779(3) 1.337(4) 0.104(1) 2 0.31816(3) 1.333(1) 0.104(1)T11 3 1 0.72971(3) 1.336(5) 0.104(1) 3 0.58862(1) 1.334(2) 0.104(1)(4 , ) 2 0.72971(4) 1.336(5) 0.104(1) 2 0.30280(4) 1.334(1) 0.104(1)Archimedean lattices. The percolation threshold of infinitelattices [ p ( ∞ ) c = lim L →∞ p c ( L )] is determined by the finite-size-scaling: [ p c ( L ) − p ( ∞ ) c ] ∼ L − a . The percolation thresh-old estimate for a finite lattice p c ( L ) is determined by theprobabilities of initial filling ( p ) that give the maximumvalues of dP ∞ ( p, L ) /dp , M (cid:48) ( p, L ), [ P w ( p, L ) − P w ( p, L )], dP w ( p, L ) /dp , and dP w ( p, L ) /dp . The fitting parame-ter a depend on lattice structure, the physical quantitymeasured, and percolation type [40]. Figure 3 confirmsthe scaling behavior very well. In the case of the BP of m = 3, however, deviation from the scaling is large in smalllattices for dP ∞ ( p, L ) /dp and M (cid:48) ( p, L ), and so results oflattices smaller than L = 100 were excluded in the fitting.This kind of deviation is also observed in BP of m = 3 inthe other kinds of lattices. The percolation threshold canalso be found by the value of p at the crossing points of P w ( p, L ) and P w ( p, L ) with various linear size ( L ) [28],as shown in Fig. 2. We ruled out the data from small lat-tices to reduce possible error from the finite-size-effect. Allthe values of the percolation threshold obtained by thesemethods were consistent with each other within error bars.The maximum of the probability of percolation only in onedirection, P w ( p, L ) − P w ( p, L ), which has negligible finite- size-effect, gives the most accurate percolation threshold.The final estimate of the percolation threshold was ob-tained by taking an average. Table 1 shows the perco-lation threshold of all continuous transitions; cases withthe first-order transition (the BP with m > m c and DPwith k < k c = ∆ + 1 − m c ) are omitted in the table.Note that m c = ∆ / m c = (∆ + 1) / m c = 3.The percolation threshold results of the BP with m = 1 or m = 2 are the same as that of the CP within error bars,as is expected. We confirmed that they are also consis-tent with exact or the most precise numerical results ofthe CP [23, 32, 41, 42] within relative errors of the orderof 0 . m and k for square, triangular,and honeycomb lattices are also consistent with references[6, 16, 17], but our work is much more precise.Figure 4 shows maximum values of the derivative ofpercolation probability, dP w ( p, L ) /dp and dP w ( p, L ) /dp ,and derivative of percolation probability and strength ofthe largest cluster P ∞ ( p, L ) at the percolation threshold p ( ∞ ) c calculated in this work. They satisfy the scaling re-4ations [35, 29]:Max [ dP w ( p, L ) /dp ] ∼ L /ν , (4)Max [ dP w ( p, L ) /dp ] ∼ L /ν , (5)[ dP w ( p, L ) /dp ] p = p ( ∞ ) c ∼ L /ν , (6)[ dP w ( p, L ) /dp ] p = p ( ∞ ) c ∼ L /ν , (7)and P ∞ ( p ( ∞ ) c , L ) ∼ L − β/ν . (8)Therefore, the critical exponents can be obtained by fit-ting. In cases of the BP of m = 3, there exists smallbut systematic deviation from the scaling behavior for P ∞ ( p ( ∞ ) c , L ) in small lattices ( L < ν = 4 / β/ν = 5 /
48) within error bars, which aremuch smaller than references [10, 11, 13, 6, 15, 16, 17].Therefore, we are convinced that continuous transitions ofthe BP and DP have the same critical exponents as theCP in two dimensions.
4. Summary
We extended the Newman-Ziff algorithm of the classi-cal percolation to propose very efficient algorithms for theBP and DP models. Using these algorithms we studiedthe BP and DP in the eleven Archimedean lattices. TheBP with m ≤ m c and the DP with k ≥ (∆ + 1 − m c )have continuous percolation phase transitions. We foundthat m c = (cid:98) (∆ + 1) / (cid:99) except for the bounce lattice (T7),which has m c = (cid:98) (∆ + 1) / (cid:99) + 1. Through the finite-size-scaling, we calculated the percolation threshold andcritical exponents for the BP and DP with the continuousphase transition. We found that the critical exponents ν and β are the same as those of the CP within error bars toconclude that the BP and DP almost certainly belong tothe same universality class as the CP in two dimensions.The algorithms presented in this paper can be directlyapplied to any dimensions and graphs. Since the BP andDP models are useful both in materials on lattices and incomplex systems, studies of the BP and DP using thesenew algorithms in three-dimensional lattices and complexnetworks would be also interesting. Acknowledgments
This work was supported by GIST Research Institute(GRI) grant funded by the GIST in 2019.
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0 5 10 15 20 25 M ′ P w L = 50 L = 100 L = 200 L = 400 L = 800 L =1600 0 0.2 0.4 0.6 0.8 P w P o p Classical percolation
0 5 10 15 20 L = 50 L = 100 L = 200 L = 400 L = 800 L =1600 0 0.2 0.4 0.6 0.8 p Bootstrap percolation( m = 3)
0 40 80 120 160 L = 50 L = 100 L = 200 L = 400 L = 800 L =1600 0 0.2 0.4 0.6 0.8 p Figure 2: Strength of the largest cluster ( P ∞ ), average cluster size excluding the largest one ( M (cid:48) ), percolation probability in any direction( P w ) and in both directions ( P w ), and proportion of occupied sites ( P o ) as a function of initial filling probability ( p ) for the DP ( k = 4), CP,and BP ( m = 3) in the triangular lattice (T1) for various linear size L . Vertical dotted lines indicate our estimates of percolation thresholds( p k DP , p CP , and p m BP ). Diagonal dotted straight lines in the lowest row represent P o = p . DP ( k = 4) p c ( L ) L Max[ dp ∞ / dp ]Max[ M ′ ]Max[ P w − P w ] Max[ dP w / dp ]Max[ dP w / dp ] 0.42 0.44 0.46 0.48 0.5 0 0.005 0.01 0.015 0.02 0.025 0.03 CP L Max[ dp ∞ / dp ]Max[ M ′ ]Max[ P w − P w ] Max[ dP w / dp ]Max[ dP w / dp ] 0.58 0.6 0.62 0 0.005 0.01 0.015 0.02 0.025 0.03 BP ( m = 3) L Max[ dp ∞ / dp ]Max[ M ′ ]Max[ P w − P w ] Max[ dP w / dp ]Max[ dP w / dp ] Figure 3: The percolation threshold estimate of finite systems p c ( L ) as a function of linear size L for DP ( k = 4), CP, and BP ( m = 3) in thetriangular lattice. They were determined by the probabilities of initial filling ( p ) that give the maximum of dP ∞ /dp , M (cid:48) , d ( P w − P w ) /dp , dP w /dp , and dP w /dp . Solid lines are from fitting of [ p c ( L ) − p ( ∞ ) c ] ∼ L − a . In the right panel, small systems ( L < dP ∞ /dp ] and Max[ M (cid:48) ].
10 20 40 80 160 320
DP ( k = 4) Max[ dP w / dp ]Max[ dP w / dp ] 10 20 40 80 160 320 CP Max[ dP w / dp ]Max[ dP w / dp ] 10 20 40 80 160 320 BP ( m = 3) Max[ dP w / dp ]Max[ dP w / dp ] 10 20 40 80 160 320 DP ( k = 4) [ dP w / dp ] p = p DP [ dP w / dp ] p = p DP
10 20 40 80 160 320 CP [ dP w / dp ] p = p CP [ dP w / dp ] p = p CP
10 20 40 80 160 320
BP ( m = 3) [ dP w / dp ] p = p BP [ dP w / dp ] p = p BP DP ( k = 4) L P ∞ ( p = p DP ) 0.2 0.3 0.4 0.5 50 100 200 400 800 1600 CP L P ∞ ( p = p CP ) 0.2 0.3 0.4 0.5 50 100 200 400 800 1600 BP ( m = 3) L P ∞ ( p = p BP ) Figure 4: Maximum values of dP w /dp and dP w /dp in the upper panels, and values of dP w /dp , dP w /dp , and P ∞ at the percolationthreshold p ( ∞ ) c in the middle and lower panels as a function of system’s linear size L in the triangular lattice in log-log scale. Solid lines arefrom fitting. In the right lower panel, small systems ( L <100) were excluded in fitting.