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Encyclopedia of Complexity and Systems Science pp 6521–6538Cite as

Percolation Lattices, Efficient Simulation of Large

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Definition of the Subject

Percolation is a simple model of the formation of long-range connectivity in random systems. While it can be solved exactly in a few cases of branchedlattices, and while many results in two dimensions (2d) can be found exactly, most of the work in this field is intimately connected with computersimulation. Various algorithms have been developed over the years, and this article surveys some of them, especially related to cluster sizes andconnectivity, and the hull.

Introduction

Percolation was introduced by Flory in 1941 [13] for branched networks (polymers) andBroadbent and Hammersley in 1957 for lattice networks, and its study by computer simulation began just a few years later [77]. The overall development of the percolation field in the ensuing years has been intimately connected withadvances in simulation and computer algorithms. Specific computer algorithms allow optimal simulation of different aspects of the percolating system, andthe results of these...

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Abbreviations

Hull:

The boundary of a percolation cluster, either internal or external.

Accessible hull:

The hull with pinched off “fjords” removed.

Hull-generating walk:

A way to generate the hull of a percolation cluster by a type of kinetic self-avoiding walk.

Queue:

A computer list construct in which the events are stored in such that the first in is the first out (also called “FIFO” or “breadth-first searching”).

Stack:

A computer list construct in which the events are stored in such a way that the last in is the first out (also called “LIFO” or “depth-first searching”).

Recursion:

A programming method in which a procedure calls itself, creating new local variables each time.

Tree:

A data structure in which points are connected in a tree-like structure with branches but no loops.

Stochastic Loewner evolution (SLE):

A theoretical way to study conformally invariant random curves, including the hulls of percolation clusters, through a transformation of simple Brownian motion. Also called Schramm–Loewner Evolution.

Leath algorithm:

A technique where individual percolation clusters are “grown” from a seed by an epidemic type of process.

Hoshen–Kopelman algorithm:

A technique where a lattice (in 2d) is scanned one row at a time, and clusters are identified using information from the previous row only.

Newman–Ziff algorithm:

A way to efficiently generate microcanonical (fixed occupancy) states and from them to study all canonical (fixed p) states.

Bibliography

Primary Literature

  1. Aharony A Stauffer D (1997) Test of universal finite-size scaling in two-dimensional site percolation. J Phys A 30:L301–L306

    ADS  MATH  Google Scholar 

  2. Alexander S, Orbach R (1982) Density of states on fractals: fractons. J Phys Lett 43:L623

    Google Scholar 

  3. Ballesteros HG, Fernández LA, Martín-Mayor V, Muñoz Sudupe A, Parisi G, Ruiz-Lorenzo JJ (1999) Scaling corrections: site percolation and Ising model in three dimensions. J Phys A 32:1–13

    Google Scholar 

  4. Berlyand L, Wehr J (1995) The probability distribution of the percolation threshold in a large system. J Phys A 28:7127–7133

    MathSciNet  ADS  MATH  Google Scholar 

  5. Cardy JL (1992) Critical percolation in finite geometries. J Phys A 25:L201–206

    MathSciNet  ADS  MATH  Google Scholar 

  6. Cardy J, Ziff RM (2003) Exact area distribution for critical percolation, Ising and Potts model clusters. J Stat Phys 110:1–33

    MathSciNet  MATH  Google Scholar 

  7. Coniglio A (1982) Cluster structure near the percolation threshold. J Phys A 15:3829–3844

    MathSciNet  ADS  Google Scholar 

  8. de Freitas JE, Lucena LS (2000) Equivalence between the FLR time-dependent percolation model and the Newman–Ziff algorithm. Int J Mod Phys C 8:1581–1584

    Google Scholar 

  9. Deng Y, Blöte HWJ, Nienhuis B (2004) Backbone exponents of the two-dimensional q-state Potts model: A Monte Carlo investigation. Phys Rev E 69:026114

    Google Scholar 

  10. Deng Y, Blöte HWJ (2005) Monte Carlo study of the site percolation model in two and three dimensions. Phys Rev E 72:016126

    Google Scholar 

  11. Erdos P, Rényi A (1959) On random graphs. Publ Math 6:290–297

    Google Scholar 

  12. Fisher M, Essam JW (1961) Some cluster size and percolation problems. J Math Phys 2:609–619

    MathSciNet  ADS  MATH  Google Scholar 

  13. Flory P (1941) Molecular Size Distribution in Three Dimensional Polymers I Gelation. J Am Chem Soc 63:3083–3090

    Google Scholar 

  14. Frisch HL, Sonnenblick E, Vyssotsky V, Hammersley JM (1961) Critical percolation probabilities (site problem) Phys Rev 124:1023–1022

    ADS  Google Scholar 

  15. Gamsa A, Cardy J (2007) SLE in the three-state Potts model – a numerical study. J Stat Mech P08020

    Google Scholar 

  16. Gould H, Tobochnik J (1996) An Introduction to Computer Simulation Methods, 2nd edn. Addison-Wesley, Reading, p 444

    Google Scholar 

  17. Gould H, Tobochnik J, Christian W (2006) An Introduction to Computer Simulation Methods, 3nd edn. Addison-Wesley, Reading

    Google Scholar 

  18. Grassberger P (1983) Critical behavior of the general epidemic process and dynamical percolation. Math Biosci 63:157–172

    MATH  Google Scholar 

  19. Grassberger P (1986) On the hull of two-dimensional percolation clusters. J Phys A 19:2675–2677

    MathSciNet  ADS  Google Scholar 

  20. Grassberger P (1992) Spreading and backbone dimensions of 2D percolation. J Phys A 25:5475–5484

    ADS  Google Scholar 

  21. Grassberger P (1999) Conductivity exponent and backbone dimension in \( { 2-d } \) percolation. Physica A 262:251–263

    MathSciNet  Google Scholar 

  22. Grassberger P, Zhang YC (1996) Self-organized formulation of standard percolation phenomena. Physica A 224:169–179

    ADS  Google Scholar 

  23. Grossman T, Aharony A (1984) Structure and perimeters of percolation clusters. J Phys A 19:L745–L751

    Google Scholar 

  24. Gruzberg I (2006) Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory. J Phys A 39:12601–12656

    MathSciNet  ADS  MATH  Google Scholar 

  25. Havlin S, Nossal R (1984) Topological properties of percolation clusters. J Phys A 17:L427–L432

    MathSciNet  ADS  Google Scholar 

  26. Herrmann HJ, Derrida B, Vannimenus J (1984) Superconductivity exponents In: Herrmann HJ, Derrida B, Vannimenus J (eds) Two-and three-dimensional percolation. Phys Rev B 30:4080–4082

    ADS  Google Scholar 

  27. Herrmann HJ, Hong DC, Stanley HE (1984) Backbone and elastic backbone of percolation clusters obtained by the new method of burning. J Phys A 17:L261–L266

    ADS  Google Scholar 

  28. Hong DC, Havlin S, Herrmann HJ, Stanley HE (1984) Breakdown of Alexander-Orbach conjecture for percolation: Exact enumeration of random walks on percolation backbones. Phys Rev B 30:4083–4086

    ADS  Google Scholar 

  29. Hoshen J Kopelman R (1976) Percolation and cluster distribution I Cluster multiple labeling technique and critical concentration algorithm. Phys Rev B 14:3438–3445

    ADS  Google Scholar 

  30. Hovi J-P, Aharony A (1996) Scaling and universality in the spanning probability for percolation. Phys Rev E 53:235–253

    ADS  Google Scholar 

  31. Hu CK (1992) Histogram Monte Carlo renormalization-group method for percolation problems. Phys Rev B 14:6592–6595

    ADS  Google Scholar 

  32. Hu CK, Chen J-A, Izmailian S Sh, Kleban P (2000) Recent developments in the Monte Carlo approach to percolation problems. Comp Phys Comm 126:77–81

    ADS  MATH  Google Scholar 

  33. Jensen I, Ziff RM (2006) Universal amplitude ratio \( { \Gamma^-/\Gamma^+ } \) for two-dimensional percolation. Phys Rev E 74:020101R

    ADS  Google Scholar 

  34. Kleban P, Ziff RM (1998) Exact results at the two-dimensional percolation point. Phys Rev B 57:R8075–R8078

    ADS  Google Scholar 

  35. Kleban P, Simmons JJH, Ziff RM (2006) Anchored critical percolation clusters and 2D electrostatics. Phys Rev Lett 97:115702

    ADS  Google Scholar 

  36. Kunz H, Souillard B (1978) Essential singularity in the percolation model. Phys Rev Lett 40:133–135

    ADS  Google Scholar 

  37. Leath P (1976) Cluster size and boundary distribution near percolation threshold. Phys Rev B 14:5046–5055

    ADS  Google Scholar 

  38. Lobb CJ, Frank DJ (1984) Percolation conduction and the Alexander-Orbach conjecture in two dimensions. Phys Rev B 30:4090–4092

    ADS  Google Scholar 

  39. Lorenz CD, Ziff RM (1998) Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices. Phys Rev E 57:230–236

    ADS  Google Scholar 

  40. Lorenz CD, May R, Ziff RM (2000) Similarity of percolation thresholds on the hcp and fcc lattices. J Stat Phys 98:961–970

    MATH  Google Scholar 

  41. Martín-Herrero J (2004) Hybrid cluster identification. J Phys A 37:9377–9386

    Google Scholar 

  42. Moore C, Newman MEJ (2000) Exact solution of site and bond percolation on small-world networks. Phys Rev E 62:7059–7064

    ADS  Google Scholar 

  43. Moukarzel C (1998) A Fast Algorithm for Backbones. Int J Mod Phys C 9:887–895

    ADS  Google Scholar 

  44. Newman MEJ, Ziff RM (2000) Efficient Monte Carlo algorithm and high-precision results for percolation. Phys Rev Lett 85:4104–4107

    ADS  Google Scholar 

  45. Newman MEJ, Ziff RM (2001) A fast Monte Carlo algorithm for site or bond percolation. Phys Rev E 64:016706

    ADS  Google Scholar 

  46. Osterkamp D, Stauffer D, Aharony A (2003) Anomalous diffusion at percolation threshold in high dimensions of 10\( { ^{18} } \) sites. Int J Mod Phys C 14:917–924

    ADS  Google Scholar 

  47. Parviainen R (2007) Estimates of the bond percolation thresholds on the Archimedean lattices. J Phys A 40:9253–9258

    MathSciNet  ADS  MATH  Google Scholar 

  48. Paul G, Ziff RM, Stanley HE (2001) Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions. Phys Rev E 64:026115

    ADS  Google Scholar 

  49. Perlsman E, Havlin S (2002) Method to estimate critical exponents using numerical studies. Europhys Lett 58:176–181

    ADS  Google Scholar 

  50. Pike R, Stanley HE (1981) Order propagation near the percolation threshold. J Phys A 14:L169

    ADS  Google Scholar 

  51. Pinson HT (1994) Critical percolation on the torus. J Stat Phys 75:1167–1177

    MathSciNet  ADS  MATH  Google Scholar 

  52. Privman V, Hohenberg PC, Aharony A (1991) Universal Critical-Point Amplitude Relations. In: Domb C, Lebowitz JL (eds) Phase transition and critical phenomena, vol 14. Academic Press, New York

    Google Scholar 

  53. Quintanilla J, Torquato S, Ziff RM (2000) Efficient measurement of the percolation threshold for fully penetrable discs. J Phys A 33:L399–L407

    MathSciNet  ADS  MATH  Google Scholar 

  54. Quintanilla J, Ziff RM (2007) Near symmetry of percolation thresholds of fully penetrable disks with two different radii. Phys Rev E 76:051115

    ADS  Google Scholar 

  55. Rammal R, Angles d'Auriac JC, Benoit A (1984) Universality of the spectral dimension of percolation clusters. Phys Rev B 30:4087–4089

    ADS  Google Scholar 

  56. Reynolds P, Stanley HE, Klein W (1980) Large-cell Monte Carlo renormalization group for percolation. Phys Rev B 21:1223–1245

    ADS  Google Scholar 

  57. Rosso M, Gouyet JF, Sapoval B (1985) Determination of percolation probability from the use of a concentration gradient. Phys Rev B 32:6063–6054

    ADS  Google Scholar 

  58. Rozenfeld HD, ben-Avraham D (2007) Percolation in hierarchical scale-free nets. Phys Rev E 75:061102

    ADS  Google Scholar 

  59. Saleur H, Duplantier B (1987) Exact determination of the percolation hull exponent in two dimensions. Phys Rev Lett 58:2325–2328

    MathSciNet  ADS  Google Scholar 

  60. Sapoval B, Rosso M, Gouyet J-F (1985) The fractal nature of a diffusion front and relation to percolation. J Phys Lett Paris 46:L149

    Google Scholar 

  61. Schramm O (1999) Scaling limits of loop-erased random walks and uniform spanning trees. Israel J Math 118:221–288

    MathSciNet  Google Scholar 

  62. Scullard CR, Ziff RM (2006) Predictions of bond percolation thresholds for the kagomé and Archimedean (3, 122) lattices. Phys Rev E 73:045102(R)

    ADS  Google Scholar 

  63. Scullard CR, Ziff RM (2008) Critical surfaces of general bond percolation problems. Phys Rev Lett 100:185701

    ADS  Google Scholar 

  64. Shchur LN (2000) Incipient Spanning Clusters in Square and Cubic Percolation. In: Landau DP, Lewis SP, Schuettler HB (eds) Springer Proceedings in Physics, vol 85. Springer, Berlin

    Google Scholar 

  65. Simmons JJH, Kleban P, Ziff RM (2007) Exact factorization of correlation functions in 2-D critical percolation. 76:041106

    Google Scholar 

  66. Simmons JJH, Kleban P, Ziff RM (2007) Percolation crossing formulas and conformal field theory. J Phys A 40:F771–F784

    MathSciNet  ADS  MATH  Google Scholar 

  67. Smirnov S, Werner W (2001) Critical exponents for two-dimensional percolation. Math Res Lett 8:729–744

    MathSciNet  MATH  Google Scholar 

  68. Stauffer D, Aharony A (1994) An Introduction to Percolation Theory, revised 2nd edn. Taylor and Francis, London

    Google Scholar 

  69. Suding PN, Ziff RM (1999) Site percolation thresholds for Archimedean lattices. Phys Rev E 60:295–283

    ADS  Google Scholar 

  70. Sykes MF, Essam JW (1964) Exact critical percolation probabilities for site and bond problems in two dimensions. J Math Phys 5:1117–1127

    MathSciNet  ADS  Google Scholar 

  71. Tarjan T (1972) Depth-first search and linear graph algorithms. SIAM J Comput 1:146–160

    MathSciNet  MATH  Google Scholar 

  72. Temperley HNV, Lieb EH (1971) Relations between the ‘percolation’and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem. Proc R Soc London A 322:251–280

    MathSciNet  ADS  MATH  Google Scholar 

  73. Tiggemann D (2001) Simulation of percolation on massively-parallel computers. Int J Mod Phys C 12:871

    ADS  Google Scholar 

  74. Voigt CA, Ziff RM (1997) Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model. Phys Rev E 56:R6241–R6244

    ADS  Google Scholar 

  75. Vollmayr H (1993) Cluster hull algorithms for large systems with small memory requirement. J Stat Phys 74:919–927

    ADS  Google Scholar 

  76. Voss RF (1984) The fractal dimension of percolation cluster hulls. J Phys A 17:L373–L377

    MathSciNet  ADS  Google Scholar 

  77. Vyssotsky VA, Gordon SB, Frisch HL, Hammersley JM (1961) Critical Percolation Probabilities (Bond Problem). Phys Rev 123:1566–1567

    MathSciNet  ADS  Google Scholar 

  78. Weinrib A, Trugman SA (1985) A new kinetic walk and percolation perimeters. Phys Rev B 31:2993–2997

    ADS  Google Scholar 

  79. Wierman JC(1984) A bond percolation critical probability determination based on the star-triangle transformation. J Phys A 17:1525–1530

    MathSciNet  ADS  Google Scholar 

  80. Wu FY (1979) Critical point of planar Potts models. J Phys C 12:L645–L650

    ADS  Google Scholar 

  81. Zabolitzky JG (1984) Monte Carlo evidence against the Alexander-Orbach conjecture for percolation conductivity. Phys Rev B 30:4077–4079

    ADS  Google Scholar 

  82. Ziff RM (1986) Test of scaling exponents for percolation cluster perimeters. Phys Rev Lett 56:545–548

    ADS  Google Scholar 

  83. Ziff RM (1992) Spanning probability in 2D percolation. Phys Rev Lett 69:2670–2674

    ADS  Google Scholar 

  84. Ziff RM (1994) Reply to Comment on Spanning probability in 2D percolation. Phys Rev Lett 72:1942

    ADS  Google Scholar 

  85. Ziff RM (1995) Proof of crossing formula for 2D percolation. J Phys A 28:6479–6480

    MathSciNet  ADS  MATH  Google Scholar 

  86. Ziff RM (1996) Effective boundary extrapolation length to account for finite-size effects in the percolation crossing function. Phys Rev E 54:2547–2554

    ADS  Google Scholar 

  87. Ziff RM (1998) Four-tap shift-register-sequence random-number generators. Comput Phys 12:385–392

    ADS  Google Scholar 

  88. Ziff RM (2004) Enclosed area distribution in percolation. Talk presented at StatPhys22. arXiv:cond-mat/0510633

    Google Scholar 

  89. Ziff RM, Cummings PT, Stell G (1984) Generation of percolation cluster perimeters by a random walk. J Phys A 17:3009–3017

    ADS  Google Scholar 

  90. Ziff RM, Sapoval B (1986) The efficient determination of the percolation threshold by a frontier-generating walk in a gradient. J Phys A 19:L1169–1172

    ADS  Google Scholar 

  91. Ziff RM, Suding PN (1997) Determination of the bond percolation threshold for the kagomé lattice. J Phys A 30:5351–5359

    MathSciNet  ADS  MATH  Google Scholar 

  92. Ziff RM, Finch SR, Adamchik VS (1997) Universality of finite-size corrections to the number of critical percolation clusters. Phys Rev Lett 79:3447–3450

    ADS  Google Scholar 

  93. Ziff RM, Lorenz CD, Kleban P (1999) Shape-dependent universality in percolation. Physica A 266:17–26

    Google Scholar 

  94. Ziff RM, Newman MEJ (2002) Convergence of threshold estimates for two-dimensional percolation. Phys Rev E 66:016129

    MathSciNet  ADS  Google Scholar 

  95. Ziff RM, Scullard CR (2006) Exact bond percolation thresholds in two dimensions. J Phys A 39:15083–15090

    MathSciNet  ADS  MATH  Google Scholar 

Books and Reviews

  1. Bollobás B, Riordan O (2006) Percolation. Cambridge University Press, Cambridge

    Google Scholar 

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Acknowledgments

This work was supported in part by the National Science Foundation under grant no. DMS-0553487. Comments by D. Stauffer andP. Kleban were highly appreciated.

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  1. Department of Chemical Engineering, University of Michigan, Ann Arbor, USA

    Robert M. Ziff

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Ziff, R.M. (2009). Percolation Lattices, Efficient Simulation of Large. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_386

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