Percolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone
PPercolation thresholds on triangular lattice for neighbourhoods containing sites up-tothe fifth coordination zone
Krzysztof Malarz ∗ AGH University of Science and Technology,Faculty of Physics and Applied Computer Science,al. Mickiewicza 30, 30-059 Krak´ow, Poland (Dated: February 22, 2021)We determine thresholds p c for random-site percolation on a triangular lattice for all availableneighborhoods containing sites from the first to the fifth coordination zones, including their complexcombinations. There are 31 distinct neighbourhoods. The dependence of the value of the percolationthresholds p c on the coordination number z are tested against various theoretical predictions. Thenewly proposed single scalar index ξ = (cid:80) i z i r i /ζ i (depending on the coordination zone number ζ ,the neighbourhood coordination number z and the square-distance r to sites in ζ -th coordinationzone from the central site) allows to differentiate among various neighbourhoods and relate p c to ξ .The thresholds roughly follow a power law p c ∝ ξ − γ with γ ≈ . Keywords: random site percolation; triangular lattice; complex and extended neighborhoods; Newman–Ziffalgorithm; Bastas et al. method; finite size scaling hypothesis; analytical formulas for percolation thresholds
I. INTRODUCTION
The concepts of site and bond percolation [1, 2] intro-duced in middle fifties [3, 4] and since then have beenapplied in various fields of science ranging from agricul-ture [5] via studies of polymer composites [6], materi-als science [7], forest fires [8], oil and gas exploration[9], quantifying urban areas [10], Bitcoins transfer [11],diseases propagation [12] to transportation networks [13](see Refs. 14 and 15 for reviews).Percolation is an example of phenomenon where a geo-metrical phase transition (on d -dimensional lattice) takesplace. The critical parameter (called percolation thresh-old p c [16]) separates two phases: one for a low occupa-tion probability p < p c and the other for p > p c . In thelow- p phase the system behaves as an insulator (withoutconnectivity path leading between system boundaries)while for the high- p phase there is a giant componentspanning the system and connecting opposite boundaries;in effect the system behaves as a conductor, when onerefers to the electric analogy.The percolation thresholds were initially estimated fornearest neighbour interactions [17, 18] but later also com-plex (or extended) neighbourhoods were studied for 2D(square [19–25], triangular [19, 20, 26, 27], honeycomb[19]), 3D (simple cubic [25, 28, 29]) and 4D (simple hyper-cubic [30]) lattices.Very recently, we have computed percolation thresh-olds for random site percolation on triangular lattice withcomplex neighbourhoods with hexagonal symmetry [27].Here we supplement these results with 31 percolationthresholds estimations for all neighbourhoods on trian-gular lattice containing sites from the first, the second,the third, the fourth and the fifth coordination zones (see ∗ Figure 1). Some of these neighbourhoods—those con-taining sites from the fifth coordination zone—are pre-sented in Figure A1 in Appendix A. The lattice namesfollow convention proposed in Ref. 25 reflecting latticetopology (here tr , i.e. triangular lattice) and numericalstring specifying the coordination zones ζ , where sitesconstituting the neighbourhood come from.Additionally—for triangular lattice and complexneighbourhoods—we test the dependence of p c on thecoordination number z , following the idea of Ref. 25.However, instead of values of the percolation thresholdsfor selected (mainly compact) neighbourhoods, we usethe mean values ¯ p c of percolation thresholds p c averagedover all available neighbourhoods with given coordina-tion number z . Unfortunately, values of ¯ p c do not followany of dependencies proposed in Ref. 25.Finally, we propose a scalar quantity ξ which may behelpful for differentiating among various neighbourhoods.The quantity is based on the coordination zone ζ , thesites number z and the sites distances r to the centralsite in the neighbourhood. The dependency of p c on thisnewly proposed index ξ follows roughly a power law withan exponent close to − . II. METHODS
In order to evaluate the percolation thresholds p c wefollow the scheme applied previously in Ref. 27. Namely,we combine Newman and Ziff [31], Bastas et al. [32] al-gorithms with finite size corrections [1, 33] to estimate p c : 1. Based on the Newman and Ziff [31] algorithm wecalculate the size of the largest cluster (cid:104)S max (cid:105) onthe number of occupied sites n (Figure 2a). Thebrackets (cid:104)· · · (cid:105) represent averaging over R = 10 lattice realizations. Applying a Gaussian approxi- a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b (a) tr -1: ζ = 1 r = 1, z = 6 (b) tr -2: ζ = 2 r = 3, z = 6 (c) tr -3: ζ = 3 r = 4, z = 6 (d) tr -4: ζ = 4 r = 7, z = 12 (e) tr -5: ζ = 5 r = 9, z = 6 FIG. 1: (Color online). Basic neighbourhoods corresponding to subsequent coordination zones ζ = 1 , · · · , r stands for the Euclidean distance of black sites to the central one (in red) and z indicates the number of sites in the neighbourhood. Examples of fifteen neighborhoods containing thenext-next-next-next-nearest neighbors (with sites from the 5-th coordination zone) on triangular lattice arepresented in Figure A1 in Appendix A.mation to the he Bernoulli distribution S max ( n ; N, p ) = (cid:18) Nn (cid:19) p n (1 − p ) N − n (1)one can calculate S max for different values of theoccupation probability p using the Gauss function: G ( n ; µ, σ ) = 1 √ πσ exp (cid:18) − ( n − µ ) σ (cid:19) , (2)with the expected value µ = pN and variance σ = p (1 − p ) N . The dependence S max ( n ; N, p ) yields aprobability of belonging to the largest cluster P max = (cid:104)S max (cid:105) /N, (3)for N = L and L = 64, 128, 256, 512, 1024, 2048and 4096 sites as presented in Figure 2b;2. Using Bastas et al. [32] algorithm we minimize thepair-wise difference λ ( p ) = (cid:88) i (cid:54) = j [ H ( p ; N i ) − H ( p ; N j )] (4)function (see Figure 2c) with H ( p ; L ) = L β/ν ·P max ( p ; L ) + 1 / [ L β/ν · P max ( p ; L )] [34] and with ex-ponents β = and ν = [1, p. 54]. The minimumof λ ( p ) estimates the percolation threshold ˆ p c ;3. Finally, in Figure 2d we plot the estimated val-ues of percolation thresholds ˆ p c ( L ) for differentranges of summation in Equation (4)—up to L =max( L i,j ) = 512, 1024, 2048 and 4096. Accordingto the standard finite size scaling [1, p. 77]ˆ p c ( L ) = p c + a · L − /ν , (5)where p c is the percolation threshold for an in-finitely large system. The least squares linear fitto data presented in Figure 2d predicts p c and itsuncertainty u ( p c ). III. RESULTS
In Figure 2 we present examples of results used to pre-dict the percolation thresholds p c as described in Sec-tion II.In Figure 2a the dependence of (cid:104)S max ( n ; L ) (cid:105) on thenumber of occupied sites n (normalized to the systemsize N ) are presented. The dependence for all dis-cussed neighbourhoods is presented in Figure A2 in Ap-pendix A. With the increase of the system size N = L the curves describing this dependence become steeperand steeper. For infinitely large systems the function d (cid:104)S max ( n ; L ) (cid:105) /dn becomes discontinuous at p = p c .In Figure 2b we show the dependence of the nor-malized probability of belonging to the largest cluster P max ( p ; L ) · L β/ν on the sites occupation probability p for various linear system sizes L . The curves represent-ing this dependence for all discussed neighbourhoods arepresented in Figure A3 in Appendix A. The abscissas ofthe points where curves intercept each other estimate thepercolation thresholds ˆ p c .In Figure 2c the dependence of λ ( p ) on the occupa-tion probability p is presented. The curves representingthis dependence for all discussed neighbourhoods are pre-sented in Figure A4 in Appendix A. The minima of λ ( p )give estimates of the percolation thresholds ˆ p c .In Figure 2d finite size corrections to ˆ p c accordingto Equation (5) are presented. The curves representingthis dependence for all discussed neighbourhoods are pre-sented in Figure A5 in Appendix A. The initial value ofthe linear fit function predicts the percolation thresholdvalues p c .The obtained percolation thresholds p c together withtheir uncertainties and earlier estimates are gathered inTable I. We used all lattice sizes 64 ≤ L ≤ p c (they are collected in the sixth columnof Table I), while estimations given in the fifth columnrely on systems with 128 ≤ L ≤ L = 64, have been excluded from calcula-tions). For neighbourhoods containing 30 or more siteswe were able to carry out simulations for lattices up to (a) tr -5: The mean largest cluster size (cid:104)S max (cid:105) vs. thenumber of occupied sites, normalized to the system size N h S m a x i / N n/N L = (b) tr -5: The probability of belonging to the largest cluster (cid:104)P max (cid:105) vs. the occupation probability p h P m a x i · L β / ν p L = (c) tr -5: (cid:104) λ ( p ) (cid:105) vs. the occupation probability p . . . . . . . . . h λ i p L = (d) tr -5: ˆ p c vs. L − /ν L − /ν ˆ p c FIG. 2: (Color online). Subsequent steps of the percolation threshold estimation (example for tr -5 lattice). (a) TheDependence of (cid:104)S max ( n ; L ) (cid:105) on the number of occupied sites n (normalized to the system size N ). (b) Thedependence of (cid:104)P max ( p ; L ) (cid:105) · L β/ν on the occupation probability p . (c) The dependence of (cid:104) λ ( p ) (cid:105) on the occupationprobability p . The minima give estimates of the percolation thresholds ˆ p c . (d) Finite size-scaling corrections to ˆ p c according to Equation (5). L = 2048. To check how the minimisation of pair-wisedifference function λ (4) reduces the finite size effects wepresent also values of p for which λ reaches the minimumwhen p is scanned with ∆ p = 10 − accuracy (see thefourth column of Table I). IV. DISCUSSION
In Ref. 25 several analytical formulas for the depen-dence of the percolation threshold p c on the coordinationnumber z were tested, with p c = c/ ( z + b ) , (6) p c = 1 − exp( d/z ) (7) among the others. These formulas work fine for “com-pact” neighbourhoods (for instance tr -1, tr -1,2,3 and tr -1,2,3,4,5, here).Unfortunately, for complex neighbourhoods these for-mulas must fail as the dependence p c on z is “degener-ated”, i.e. several values of p c are associated with thesame number z of sites in the neighbourhood (see Fig-ure 3a, and also Figure 4b in Ref. 24 for the square lat-tice). These degeneration is also observed for basic neigh-bourhoods what brought some brickbats [35] on the pos-sibility of existing universal formulas for the percolationthreshold (depending solely on the spatial dimension d ofthe system and the coordination number z ), as proposedin Galam and Mauger [36].To remove this degeneracy we tested formulas (6) and(7) for the mean values ¯ p c of percolation thresholds.The averaging, denoted by the bar, goes over percola-TABLE I: Estimated values of random site triangular lattice percolation thresholds p c for various complexneighbourhoods. The lattice name encodes the coordination zones ζ , to which sites in the neighbourhood belong.Also the coordination number z , the index ξ and the value of p at the minimum of λ are presented. lattice z ξ p at min (cid:104) λ (cid:105) p c p c earlier estimations(128 ≤ L ≤ ≤ L ≤ ≤ L ≤ p = 10 − tr -1,2,3,4,5 36 54.8 0.1157 40 a a a tr -2,3,4,5 30 48.8 0.1174 40 a a a tr -1,3,4,5 30 45.8 0.1215 48 a a a tr -1,2,4,5 30 46.8 0.1225 93 a a a tr -1,2,3,5 24 33.8 0.1522 59 0.152297(17) 0.152282(10) tr -3,4,5 24 39.8 0.1255 48 0.1255511(43) 0.1255483(43) tr -2,4,5 24 40.8 0.1266 22 0.126653(11) 0.1266400(40) tr -2,3,5 18 27.8 0.1616 45 0.161664(15) 0.161653(15) tr -1,4,5 24 37.8 0.1316 69 0.1316677(13) 0.13166484(66) 0.131792(58) [27] tr -1,3,5 18 24.8 0.1700 42 0.1700473(87) 0.170039(10) tr -1,2,5 18 25.8 0.1762 42 0.176263(11) 0.1762610(92) tr -4,5 18 31.8 0.1402 43 0.1402453(79) 0.1402382(92) 0.140286(5) [27] tr -3,5 12 18.8 0.1957 03 0.1956981(14) 0.1957039(24) tr -2,5 12 19.8 0.2902 68 0.290280(20) 0.290279(17) tr -1,5 12 16.8 0.2095 62 0.209563(13) 0.209561(10) tr -5 6 10.8 0.5000 00 0.5000029(40) 0.4999961(55) [27] tr -1,2,3,4 30 44 0.1358 13 a a a tr -2,3,4 24 38 0.1391 15 0.1391118(33) 0.1391117(26) tr -1,3,4 24 35 0.1443 07 0.1443064(38) 0.1443074(32) tr -1,2,4 24 36 0.1489 78 0.1489791(88) 0.1489757(74) tr -3,4 18 29 0.1519 32 0.1519532(26) 0.1519393(35) tr -2,4 18 30 0.1584 53 0.1584634(54) 0.1584620(43) tr -1,4 18 27 0.1651 88 0.165186(14) 0.165186(12) tr -4 12 21 0.1924 37 0.1924356(68) 0.1924428(50) 0.192410(43) [27] tr -1,2,3 18 23 0.2154 62 0.21546261(91) 0.2154657(17) 0.215484(19) [27], 0.215 [26] tr -2,3 12 17 0.2320 12 0.232019(23) 0.232020(20) 0.232008(38) [27] tr -1,3 12 14 0.2645 25 0.264545(25) 0.264539(21) tr -3 6 8 0.5000 24 0.500027(31) 0.500013(23) [27] tr -1,2 12 15 0.2902 67 0.290261(22) 0.290258(19) 0.295 [19] tr -2 6 9 0.4999 85 0.499987(20) 0.499978(20) [27] tr -1 6 6 0.4999 93 0.499994(17) 0.499996(14) [1, p. 17] a L ≤ tion thresholds for neighbourhoods for fixed number z .For instance ¯ p c ( z = 6) is the mean value of p c for tr -1, tr -2, tr -3, tr -5 lattices, while ¯ p c ( z = 30) is based onvalues of p c for tr -1,2,3,4, tr -1,2,4,5, tr -1,3,4,5 and tr -2,3,4,5 lattices, respectively. Additionally, we checkedthe quality of such fits for the Galam–Mauger formula[36], which for fixed network topology reduces to the fol-lowing power-law dependence p c ∝ ( z − − a . (8)Unfortunately, the formulas ¯ p c ( z ) describing the depen- dence on z are not consistent with either (6) or (7) or (8).However, we notice that the maximum value of p c for thefixed value of coordination number z follows a power lawmax z =const p c ∝ z − δ (9)with the power δ ≈ . d and z seems to be insuffi-cient to differentiate among various lattices with assumedneighbourhoods we are looking for another index ξ whichmay be useful for both: deriving a formula p c ( ξ ) and re- (a) p c ( z ) p c z (b) p c ( ξ ) ξ p c tr -2,5 FIG. 3: (Color online). Percolation thresholds for complex neighbourhoods on triangular lattice. (a) Degenerateddependence of the percolation threshold p c on the coordination number z . The maxima of p c for fixed z followEquation (9). The power is δ ≈ . p c vs. ξ for complexneighbourhoods. The exponent γ in Equation (9) is 0 . p c values for equivalents of tr -1 neighbourhoods(marked with crosses) are excluded from fitting.moving the p c -degeneracy. The attempts for such indexidentification were undertaken also in graph theory (fortopological invariants for trees [37]) and in organic chem-istry (for molecular topological index [38–40]).Here, we propose a weighted square distance r i of z i sites in the given neighbourhood which belong to ζ i -thcoordination zone ξ = (cid:88) i z i r i /ζ i . (10)The values of ξ are presented in Table I.For complex neighbourhoods the dependence p c ( ξ ) fol-lows roughly a power-law p c ∝ ξ − γ (11)with γ ≈ . p c for tr -2,5 (see Figure 3b). Please note that p c ( tr -1) = p c ( tr -2) = p c ( tr -3) = p c ( tr -5) are exactly the sameand equal to as these neighbourhoods are equivalentto those described in Ref. 27. Thus p c values for tr -2, tr -3, tr -5 neighbourhoods are excluded from fitting. V. CONCLUSIONS
Concluding, in this paper we estimated percolationthresholds p c for random site triangular lattice perco-lation and for neighborhoods containing sites from thefirst to fifth coordination zone. The estimated values ofpercolation thresholds are collected in Table I.We note that the method Bastas et al. [32] allows (atleast partially) to get rid of finite size effects. The min-ima of the pairwise difference λ function presented in Table I are consistent with a five digit accuracy with thepercolation threshold p c obtained for infinite lattice ac-cording to Equation (5). The five digit accuracy seemsto outperform by at least one order of magnitude practi-cal requirements on experimenters in any field of sciencewhere percolation theory may be applied.As the percolation thresholds p c ( z ) are multiply de-generated, we propose the weighted square distance ξ todifferentiate among various neighbourhoods. This indexseems to be effective in this respect (at least for neigh-bourhoods investigated here).Finally, the p c ( ξ ) dependence follows roughly a powerlaw (11). The deviation from this dependence is observedonly for p c estimated for tr -2,5 lattice. The obtained re-sults may be useful in further searching for the empiricalformula for percolation thresholds [41]. Appendix A: Supplementary material
In Figure A1 neighbourhoods containing sites from ζ = 5 coordination zone are presented. In Figure A2the mean largest cluster size (cid:104)S max (cid:105) vs. the number ofoccupied sites, normalized to the system size N are pre-sented. Figure A3 shows the probability of belongingto the largest cluster (cid:104) P max (cid:105) vs. occupation probability p . Sums of the pairwise difference λ vs. the occupationprobability p are presented in Figure A4. In Figure A5finite size corrections to p c are presented. [1] D. Stauffer and A. Aharony, Introduction to PercolationTheory , 2nd ed. (Taylor and Francis, London, 1994).[2] J. Wierman, “Percolation theory,” in
Wiley StatsRef:Statistics Reference Online (American Cancer Society,2014) pp. 1–9.[3] S. R. Broadbent and J. M. Hammersley, “Percolationprocesses: I. Crystals and mazes,” Mathematical Pro-ceedings of the Cambridge Philosophical Society , 629–641 (1957).[4] J. M. Hammersley, “Percolation processes: II. Theconnective constant,” Mathematical Proceedings of theCambridge Philosophical Society , 642–645 (1957).[5] J. E. Ram´ırez, C. Pajares, M. I. Mart´ınez,R. Rodr´ıguez Fern´andez, E. Molina-Gayosso, J. Lozada-Lechuga, and A. Fern´andez T´ellez, “Site-bond per-colation solution to preventing the propagation of Phytophthora zoospores on plantations,” PhysicalReview E , 032301 (2020).[6] Q. Zhang, B.-Y. Zhang, B.-H. Guo, Z.-X. Guo, andJ. Yu, “High-temperature polymer conductors with self-assembled conductive pathways,” Composites Part B—Engineering , 107989 (2020).[7] L. Cheng, P. Yan, X. Yang, H. Zou, H. Yang, andH. Liang, “High conductivity, percolation behavior anddielectric relaxation of hybrid ZIF-8/CNT composites,”Journal of Alloys and Compounds , 154132 (2020).[8] K. Malarz, S. Kaczanowska, and K. Ku(cid:32)lakowski, “Areforest fires predictable?” International Journal of ModernPhysics C , 1017–1031 (2002).[9] B. Ghanbarian, F. Liang, and H.-H. Liu, “Modeling gasrelative permeability in shales and tight porous rocks,”Fuel , 117686 (2020).[10] W. Cao, L. Dong, L. Wu, and Y. Liu, “Quantifying ur-ban areas with multi-source data based on percolationtheory,” Remote Sensing of Environment , 111730(2020).[11] S. Bartolucci, F. Caccioli, and P. Vivo, “A percolationmodel for the emergence of the Bitcoin Lightning Net-work,” Scientific Reports , 4488 (2020).[12] R. M. Ziff, “Percolation and the pandemic,” Physica A:Statistical Mechanics and its Applications , 125723(2021).[13] S. Dong, A. Mostafizi, H. Wang, J. Gao, and X. Li,“Measuring the topological robustness of transportationnetworks to disaster-induced failures: A percolation ap-proach,” Journal of Infrastructure Systems , 04020009(2020).[14] M. Li, R.-R. Liu, L. L¨u, M.-B. Hu, S. Xu,and Y.-C. Zhang, “Percolation on complex networks:Theory and application,” Physics Reports (2021),10.1016/j.physrep.2020.12.003.[15] A. A. Saberi, “Recent advances in percolation theory andits applications,” Physics Reports , 1–32 (2015).[16] H. L. Frisch, E. Sonnenblick, V. A. Vyssotsky, andJ. M. Hammersley, “Critical percolation probabilities(site problem),” Physical Review , 1021–1022 (1961).[17] P. Dean, “A new Monte Carlo method for percolationproblems on a lattice,” Mathematical Proceedings of theCambridge Philosophical Society , 397–410 (1963).[18] P. Dean and N. F. Bird, “Monte Carlo estimates of crit-ical percolation probabilities,” Mathematical Proceed- ings of the Cambridge Philosophical Society , 477–479(1967).[19] N. W. Dalton, C. Domb, and M. F. Sykes, “Dependenceof critical concentration of a dilute ferromagnet on therange of interaction,” Proceedings of the Physical Society , 496–498 (1964).[20] C. Domb and N. W. Dalton, “Crystal statistics with long-range forces: I. The equivalent neighbour model,” Pro-ceedings of the Physical Society , 859–871 (1966).[21] M. Gouker and F. Family, “Evidence for classical criticalbehavior in long-range site percolation,” Physical ReviewB , 1449–1452 (1983).[22] K. Malarz and S. Galam, “Square-lattice site percolationat increasing ranges of neighbor bonds,” Physical ReviewE , 016125 (2005).[23] S. Galam and K. Malarz, “Restoring site percolation ondamaged square lattices,” Physical Review E , 027103(2005).[24] M. Majewski and K. Malarz, “Square lattice site per-colation thresholds for complex neighbourhoods,” ActaPhysica Polonica B , 2191–2199 (2007).[25] Z. Xun, D. Hao, and R. M. Ziff, “Site percolation onsquare and simple cubic lattices with extended neighbor-hoods and their continuum limit,” Physical Review E , 022126 (2021).[26] C. d’Iribarne, M. Rasigni, and G. Rasigni, “From latticelong-range percolation to the continuum one,” PhysicsLetters A , 65–69 (1999).[27] K. Malarz, “Site percolation thresholds on triangular lat-tice with complex neighborhoods,” Chaos , 123123(2020).[28] (cid:32)L. Kurzawski and K. Malarz, “Simple cubic random-sitepercolation thresholds for complex neighbourhoods,” Re-ports on Mathematical Physics , 163–169 (2012).[29] K. Malarz, “Simple cubic random-site percolation thresh-olds for neighborhoods containing fourth-nearest neigh-bors,” Physical Review E , 043301 (2015).[30] M. Kotwica, P. Gronek, and K. Malarz, “Efficient spacevirtualisation for Hoshen–Kopelman algorithm,” Interna-tional Journal of Modern Physics C , 1950055 (2019).[31] M. E. J. Newman and R. M. Ziff, “Fast Monte Carloalgorithm for site or bond percolation,” Physical ReviewE , 016706 (2001).[32] N. Bastas, K. Kosmidis, P. Giazitzidis, and M. Mara-gakis, “Method for estimating critical exponents in per-colation processes with low sampling,” Physical ReviewE , 062101 (2014).[33] V. Privman, “Finite-size scaling theory,” in Finite sizescaling and numerical simulation of statistical systems ,edited by V. Privman (World Scientific, Singapore, 1990)pp. 1–98.[34] N. Bastas, K. Kosmidis, and P. Argyrakis, “Explosivesite percolation and finite-size hysteresis,” Physical Re-view E , 066112 (2011).[35] S. C. van der Marck, “Universal formulas for percolationthresholds — Comment,” Physical Review E , 1228–1229 (1997).[36] S. Galam and A. Mauger, “Universal formulas for per-colation thresholds,” Physical Review E , 2177–2181(1996). [37] S. Piec, K. Malarz, and K. Ku(cid:32)lakowski, “How to counttrees?” International Journal of Modern Physics C ,1527–1534 (2005).[38] I. Gutman, “Selected properties of the Schultz moleculartopological index,” Journal of Chemical Information andComputer Sciences , 1087–1089 (1994).[39] H. P. Schultz, “Topological organic chemistry. 1. Graphtheory and topological indices of alkanes,” Journal ofChemical Information and Computer Sciences , 227–228 (1989). [40] H. Wiener, “Structural determination of paraffin boilingpoints,” Journal of the American Chemical Society ,17–20 (1947).[41] W. Lebrecht, P.M. Centres, and A.J. Ramirez-Pastor,“Empirical formula for site and bond percolation thresh-olds on Archimedean and 2-uniform lattices,” Physica A:Statistical Mechanics and its Applications , 125802(2021). (a) tr -1,5 (b) tr -2,5 (c) tr -3,5 (d) tr -4,5 (e) tr -1,2,5(f) tr -1,3,5 (g) tr -1,4,5 (h) tr -2,3,5 (i) tr -2,4,5 (j) tr -3,4,5(k) tr -1,2,3,5 (l) tr -1,2,4,5 (m) tr -1,3,4,5 (n) tr -2,3,4,5 (o) tr -1,2,3,4,5 FIG. A1: Examples of neighborhoods containing the next-next-next-next-nearest neighbors on triangular lattice. (a) tr -1,2,3,4,5 L = S m a x / N n/N (b) tr -2,3,4,5 L = S m a x / N n/N (c) tr -1,3,4,5 L = S m a x / N n/N (d) tr -1,2,4,5 L = S m a x / N n/N (e) tr -1,2,3,5 L = S m a x / N n/N (f) tr -3,4,5 L = S m a x / N n/N (g) tr -2,4,5 L = S m a x / N n/N (h) tr -2,3,5 L = S m a x / N n/N (i) tr -1,4,5 L = S m a x / N n/N (j) tr -1,3,5 L = S m a x / N n/N (k) tr -1,2,5 L = S m a x / N n/N (l) tr -4,5 L = S m a x / N n/N (m) tr -3,5 L = S m a x / N n/N (n) tr -2,5 L = S m a x / N n/N (o) tr -1,5 L = S m a x / N n/N (p) tr -5 L = S m a x / N n/N (q) tr -1,2,3,4 L = S m a x / N n/N (r) tr -2,3,4 L = S m a x / N n/N (s) tr -1,3,4 L = S m a x / N n/N (t) tr -1,2,4 L = S m a x / N n/N (u) tr -3,4 L = S m a x / N n/N (v) tr -2,4 L = S m a x / N n/N (w) tr -1,4 L = S m a x / N n/N (x) tr -4 L = S m a x / N n/N (y) tr -1,2,3 L = S m a x / N n/N (z) tr -2,3 L = S m a x / N n/N (aa) tr -1,3 L = S m a x / N n/N (bb) tr -3 L = S m a x / N n/N (cc) tr -1,2 L = S m a x / N n/N (dd) tr -2 L = S m a x / N n/N (ee) tr -1 L = S m a x / N n/N FIG. A2: Average the largest cluster size (cid:104)S max (cid:105) vs. number of occupied sites, both normalized to the system size N (a) tr -1,2,3,4,5 P m a x · L β / ν p L = (b) tr -2,3,4,5 P m a x · L β / ν p L = (c) tr -1,3,4,5 P m a x · L β / ν p L = (d) tr -1,2,4,5 P m a x · L β / ν p L = (e) tr -1,2,3,5 P m a x · L β / ν p L = (f) tr -3,4,5 P m a x · L β / ν p L = (g) tr -2,4,5 P m a x · L β / ν p L = (h) tr -2,3,5 P m a x · L β / ν p L = (i) tr -1,4,5 P m a x · L β / ν p L = (j) tr -1,3,5 P m a x · L β / ν p L = (k) tr -1,2,5 P m a x · L β / ν p L = (l) tr -4,5 P m a x · L β / ν p L = (m) tr -3,5 P m a x · L β / ν p L = (n) tr -2,5 P m a x · L β / ν p L = (o) tr -1,5 P m a x · L β / ν p L = (p) tr -5 P m a x · L β / ν p L = (q) tr -1,2,3,4 P m a x · L β / ν p L = (r) tr -2,3,4 P m a x · L β / ν p L = (s) tr -1,3,4 P m a x · L β / ν p L = (t) tr -1,2,4 P m a x · L β / ν p L = (u) tr -3,4 P m a x · L β / ν p L = (v) tr -2,4 P m a x · L β / ν p L = (w) tr -1,4 P m a x · L β / ν p L = (x) tr -4 P m a x · L β / ν p L = (y) tr -1,2,3 P m a x · L β / ν p L = (z) tr -2,3 P m a x · L β / ν p L = (aa) tr -1,3 P m a x · L β / ν p L = (bb) tr -3 P m a x · L β / ν p L = (cc) tr -1,2 P m a x · L β / ν p L = (dd) tr -2 P m a x · L β / ν p L = (ee) tr -1 P m a x · L β / ν p L = FIG. A3: Probability of belonging to the largest cluster (cid:104)P max (cid:105) vs. occupation probability p (a) tr -1,2,3,4,5 . . . . . . . . . λ p L = (b) tr -2,3,4,5 . . . . . . . . . λ p L = (c) tr -1,3,4,5 . . . . . . . . . λ p L = (d) tr -1,2,4,5 . . . . . . . . . λ p L = (e) tr -1,2,3,5 . . . . . . . . . λ p L = (f) tr -3,4,5 . . . . . . . . . λ p L = (g) tr -2,4,5 . . . . . . . . . λ p L = (h) tr -2,3,5 . . . . . . . . . λ p L = (i) tr -1,4,5 . . . . . . . . . λ p L = (j) tr -1,3,5 . . . . . . . . . λ p L = (k) tr -1,2,5 . . . . . . . . . λ p L = (l) tr -4,5 . . . . . . . . . λ p L = (m) tr -3,5 . . . . . . . . . λ p L = (n) tr -2,5 . . . . . . . . . λ p L = (o) tr -1,5 . . . . . . . . . λ p L = (p) tr -5 . . . . . . . . . λ p L = (q) tr -1,2,3,4 . . . . . . . . . λ p L = (r) tr -2,3,4 . . . . . . . . . λ p L = (s) tr -1,3,4 . . . . . . . . . λ p L = (t) tr -1,2,4 . . . . . . . . . λ p L = (u) tr -3,4 . . . . . . . . . λ p L = (v) tr -2,4 . . . . . . . . . λ p L = (w) tr -1,4 . . . . . . . . . λ p L = (x) tr -4 . . . . . . . . . λ p L = (y) tr -1,2,3 . . . . . . . . . λ p L = (z) tr -2,3 . . . . . . . . . λ p L = (aa) tr -1,3 . . . . . . . . . λ p L = (bb) tr -3 . . . . . . . . . λ p L = (cc) tr -1,2 . . . . . . . . . λ p L = (dd) tr -2 . . . . . . . . . λ p L = (ee) tr -1 . . . . . . . . . λ p L = FIG. A4: Sum of pairwise differences (cid:104) λ (cid:105) vs. occupation probability p (a) tr -1,2,3,4,5 ˆ p c L − /ν (b) tr -2,3,4,5 ˆ p c L − /ν (c) tr -1,3,4,5 ˆ p c L − /ν (d) tr -1,2,4,5 ˆ p c L − /ν (e) tr -1,2,3,5 ˆ p c L − /ν (f) tr -3,4,5 ˆ p c L − /ν (g) tr -2,4,5 ˆ p c L − /ν (h) tr -2,3,5 ˆ p c L − /ν (i) tr -1,4,5 ˆ p c L − /ν (j) tr -1,3,5 ˆ p c L − /ν (k) tr -1,2,5 ˆ p c L − /ν (l) tr -4,5 ˆ p c L − /ν (m) tr -3,5 ˆ p c L − /ν (n) tr -2,5 ˆ p c L − /ν (o) tr -1,5 ˆ p c L − /ν (p) tr -5 ˆ p c L − /ν (q) tr -1,2,3,4 ˆ p c L − /ν (r) tr -2,3,4 ˆ p c L − /ν (s) tr -1,3,4 ˆ p c L − /ν (t) tr -1,2,4 ˆ p c L − /ν (u) tr -3,4 ˆ p c L − /ν (v) tr -2,4 ˆ p c L − /ν (w) tr -1,4 ˆ p c L − /ν (x) tr -4 ˆ p c L − /ν (y) tr -1,2,3 ˆ p c L − /ν (z) tr -2,3 ˆ p c L − /ν (aa) tr -1,3 ˆ p c L − /ν (bb) tr -3 ˆ p c L − /ν (cc) tr -1,2 ˆ p c L − /ν (dd) tr -2 ˆ p c L − /ν (ee) tr -1 ˆ p c L − /ν FIG. A5: Finite size effect corrections to p cc