Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Correlated Percolation

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_104-3

Definition of the Subject

Cluster concepts have been extremely useful in elucidating many problems in physics. Percolation theory provides a generic framework to study the behavior of the cluster distribution. In most cases, the theory predicts a geometrical transition at the percolation threshold, characterized in the percolative phase by the presence of a spanning cluster, which becomes infinite in the thermodynamic limit. Standard percolation usually deals with the problem when the constitutive elements of the clusters are randomly distributed. However, correlations cannot always be neglected. In this case, correlated percolation is the appropriate theory to study such systems. The origin of correlated percolation could be dated back to 1937 when Mayer (1937; Mayer and Ackermann 1937; Mayer and Harrison 1938; Mayer and Mayer 1940) proposed a theory to describe the condensation from a gas to a liquid in terms of mathematical clusters (for a review of cluster theory in simple fluids,...

Keywords

Fractal Dimension Ising Model Critical Exponent Coexistence Curve Infinite Cluster 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Bibliography

Primary Literature

  1. Abete T, de Candia A, Lairez D, Coniglio A (2004) Percolation Model for Enzyme Gel Degradation. Phys Rev Lett 93:228301ADSCrossRefGoogle Scholar
  2. Aharony A, Gefen Y, Kapitulnik A (1984) Scaling at the percolation threshold above six dimensions. J Phys A 17:1197MathSciNetCrossRefGoogle Scholar
  3. Aizenman M (1997) On the number of incipient spanning clusters. Nucl Phys B 485:551ADSMathSciNetMATHCrossRefGoogle Scholar
  4. Alexander S, Grest GS, Nakanishi H, Witten TA (1984) Branched polymer approach to the structure of lattice animals and percolation clusters. J Phys A 17:L185ADSMathSciNetCrossRefGoogle Scholar
  5. Amitrano C, di Liberto F, Figari R, Peruggi F (1983) A-B droplets for a two-dimensional antiferromagnetic Ising model in external field H. J Phys A Math Gen 16:3925ADSCrossRefGoogle Scholar
  6. Balog I, Uzelac K (2007) Invaded cluster algorithm for a tricritical point in a diluted Potts model. Phys Rev E 76:011103ADSCrossRefGoogle Scholar
  7. Bastiaansen PJM, Knops HJF (1997) Correlated percolation and the correlated resistor network. J Phys A Math Gen 30:1791ADSMathSciNetMATHCrossRefGoogle Scholar
  8. Bialas P, Blanchard P, Fortunato S, Gandolfo D, Satz H (2000) Percolation and magnetization in the continuous spin Ising model. Nucl Phys B 583:368ADSCrossRefGoogle Scholar
  9. Binder K (1976) “Clusters” in the Ising model, metastable states and essential singularity. Ann Phys NY 98:390ADSCrossRefGoogle Scholar
  10. Birgeneau RJ, Cowley RA, Shirane G, Guggenheim HJ (1976) Spin correlations near the percolation concentration in two dimensions. Phys Rev Lett 37:940ADSCrossRefGoogle Scholar
  11. Birgeneau RJ, Cowley RA, Shirane G, Tarvin JA, Guggenheim HJ (1980) Spin fluctuations in random magnetic-nonmagnetic two-dimensional antiferromagnets. II. Heisenberg percolation. Phys Rev B 21:317ADSCrossRefGoogle Scholar
  12. Blanchard P, Digal S, Fortunato S, Gandolfo D, Mendes T, Satz H (2000) Cluster percolation in O(n) spin models. J Phys A Math Gen 33:8603ADSMathSciNetMATHCrossRefGoogle Scholar
  13. Blote HWJ, Knops YMM, Nienhuis B (1992) Geometrical aspects of critical Ising configurations in two dimensions. Phys Rev Lett 68:3440ADSCrossRefGoogle Scholar
  14. Broderix K, Löwe H, Müller P, Zippelius A (2000) Critical dynamics of gelation. Phys Rev E 63:011510ADSCrossRefGoogle Scholar
  15. Bug ALR, Safran SA, Grest GS, Webman I (1985) Do interactions raise or lower a percolation threshold? Phys Rev Lett 55:1896ADSCrossRefGoogle Scholar
  16. Bunde A, Havlin S (1991) Percolation I. In: Bunde A, Havlin S (eds) Fractals and disordered systems. Springer, New York, pp 51–95CrossRefGoogle Scholar
  17. Campbell AI, Anderson VJ, van Duijneveldt JS, Bartlett P (2005) Dynamical arrest in attractive colloids: The effect of long-range repulsion. Phys Rev Lett 94:208301ADSCrossRefGoogle Scholar
  18. Campi X, Krivine H (2005) Zipf's law in multifragmentation. Phys Rev C 72:057602ADSCrossRefGoogle Scholar
  19. Campi X, Krivine H, Plagnol E, Sator N (2003) “Little big bang” scenario of multifragmentation. Phys Rev C 67:044610ADSCrossRefGoogle Scholar
  20. Campi X, Krivine H, Puente A (1999) On a definition of stable droplets in the lattice-gas model. Physica A 262:328ADSCrossRefGoogle Scholar
  21. Campi X, Krivine H, Sator N (2001) Percolation line of self-bound clusters in supercritical fluids. Physica A 296:24ADSMATHCrossRefGoogle Scholar
  22. Chayes JT, Chayes L, Grimmet GR, Kesten H, Schonmann R (1989) The Correlation length for the high-density phase of bernoulli percolation. Ann Probab 17:1277MathSciNetMATHCrossRefGoogle Scholar
  23. Chayes L, Coniglio A, Machta J, Shtengel K (1999) Mean-field theory for percolation models of the Ising type. J Stat Phys 94:53ADSMathSciNetMATHCrossRefGoogle Scholar
  24. Chen SH, Rouch J, Sciortino F, Tartaglia P (1994) Static and dynamic properties of water-in-oil microemulsions near the critical and percolation points. J Phys Condens Matter 6:10855ADSCrossRefGoogle Scholar
  25. Coniglio A (1975) Percolation problems and phase transitions. J Phys A 8:1773ADSCrossRefGoogle Scholar
  26. Coniglio A (1976) Some cluster-size and percolation problems for interacting spins. Phys Rev B 13:2194ADSCrossRefGoogle Scholar
  27. Coniglio A (1981) Thermal phase transition of the dilute s-state Potts and n-vector models at the percolation threshold. Phys Rev Lett 46:250ADSCrossRefGoogle Scholar
  28. Coniglio A (1982) Cluster structure near the percolation threshold. J Phys A 15:3829ADSMathSciNetCrossRefGoogle Scholar
  29. Coniglio A (1983) Proceedings of Erice school on ferromagnetic transitions. Springer, New YorkGoogle Scholar
  30. Coniglio A (1985) Finely divided matter. In: Boccara N, Daoud M (eds) Proceedings of the les Houches Winter conference. Springer, New YorkGoogle Scholar
  31. Coniglio A (1989) Fractal structure of Ising and Potts clusters: Exact results. Phys Rev Lett 62:3054ADSMathSciNetCrossRefGoogle Scholar
  32. Coniglio A (1990) Correlations in thermal and geometrical systems. In: Stanley HE, Ostrowsky W (eds) Correlation and connectivity – geometric aspects of physics, chemistry and biology, vol 188, NATO ASI series. Kluwer, DordrechtGoogle Scholar
  33. Coniglio A (2000) Geometrical approach to phase transitions in frustrated and unfrustrated systems. Physica A 281:129ADSCrossRefGoogle Scholar
  34. Coniglio A, Figari R (1983) Droplet structure in Ising and Potts models. J Phys A Math Gen 16:L535ADSCrossRefGoogle Scholar
  35. Coniglio A, Klein W (1980) Clusters and Ising critical droplets: A renormalisation group approach. J Phys A 13:2775ADSCrossRefGoogle Scholar
  36. Coniglio A, Lubensky T (1980) Epsilon expansion for correlated percolation: Applications to gels. J Phys A 13:1783ADSCrossRefGoogle Scholar
  37. Coniglio A, Peruggi F (1982) Clusters and droplets in the q-state Potts model. J Phys A 15:1873ADSCrossRefGoogle Scholar
  38. Coniglio A, Stanley HE (1984) Screening of deeply invaginated clusters and the critical behavior of the random superconducting network. Phys Rev Lett 52:1068ADSCrossRefGoogle Scholar
  39. Coniglio A, Stauffer D (1980) Fluctuations of the infinite network in percolation theory. Lett Nuovo Cimento 28:33CrossRefGoogle Scholar
  40. Coniglio A, Zia RVP (1982) Analysis of the Migdal-Kadanoff renormalisation group approach to the dilute s-state Potts model. An alternative scheme for the percolation (s to 1) limit. J Phys A Math Gen 15:L399ADSMathSciNetCrossRefGoogle Scholar
  41. Coniglio A, Nappi C, Peruggi F, Russo L (1976) Percolation and phase transitions in the Ising model. Commun Math Phys 51:315ADSMathSciNetCrossRefGoogle Scholar
  42. Coniglio A, Nappi C, Peruggi F, Russo L (1977a) Percolation points and critical point in the Ising model. J Phys A 10:205ADSMathSciNetCrossRefGoogle Scholar
  43. Coniglio A, De Angelis U, Forlani A, Lauro G (1977b) Distribution of physical clusters. J Phys A Math Gen 10:219ADSCrossRefGoogle Scholar
  44. Coniglio A, De Angelis U, Forlani A (1977c) Pair connectedness and cluster size. J Phys A Math Gen 10:1123ADSCrossRefGoogle Scholar
  45. Coniglio A, di Liberto F, Monroy G, Peruggi F (1991) Cluster approach to spin glasses and the frustrated-percolation problem. Phys Rev B 44:12605ADSCrossRefGoogle Scholar
  46. Coniglio A, Stanley HE, Klein W (1979) Site-bond correlated-percolation problem: A statistical mechanical model of polymer gelation. Phys Rev Lett 42:518ADSCrossRefGoogle Scholar
  47. Coniglio A, Stanley HE, Klein W (1982) Solvent effects on polymer gels: A statistical-mechanical model. Phys Rev B 25:6805ADSMathSciNetCrossRefGoogle Scholar
  48. Coniglio A, di Liberto F, Monroy G, Peruggi F (1989) Exact relations between droplets and thermal fluctuations in external field. J Phys A 22:L837ADSCrossRefGoogle Scholar
  49. Coniglio A, de Arcangelis L, del Gado E, Fierro A, Sator N (2004) Percolation, gelation and dynamical behaviour in colloids. J Phys Condens Matter 16:S4831ADSCrossRefGoogle Scholar
  50. Coniglio A, Abete T, de Candia A, del Gado E, Fierro A (2007) Static and dynamic heterogeneities in irreversible gels and colloidal gelation. J Phys Condens Matter 19:205103ADSCrossRefGoogle Scholar
  51. Cox MAA, Essam JW (1976) Series expansion study of the pair connectedness in site percolation models. J Phys C 9:3985ADSCrossRefGoogle Scholar
  52. de Arcangelis L (1987) Multiplicity of infinite clusters in percolation above six dimensions. J Phys A 20:3057ADSCrossRefGoogle Scholar
  53. de Candia A, del Gado E, Fierro A, Sator N, Coniglio A (2005) Colloidal gelation, percolation and structural arrest. Phys A 358:239CrossRefGoogle Scholar
  54. de Gennes PG (1975) Critical dimensionality for a special percolation problem. J Phys Paris 36:1049CrossRefGoogle Scholar
  55. de Gennes PG (1976) La percolation: Un concept unificateur. La Recherche 7:919Google Scholar
  56. de Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, IthacaGoogle Scholar
  57. del Gado E, de Arcangelis L, Coniglio A (2000) La percolation: un concept unificateur. Eur Phys J E 2:359CrossRefGoogle Scholar
  58. del Gado E, Fierro A, de Arcangelis L, Coniglio A (2004) Slow dynamics in gelation phenomena: From chemical gels to colloidal glasses. Phys Rev E 69:051103ADSCrossRefGoogle Scholar
  59. Deng Y, Guo W, Blote HWJ (2005) Percolation between vacancies in the two-dimensional Blume-Capel model. Phys Rev E 72:016101ADSCrossRefGoogle Scholar
  60. Dhar D (1999) The Abelian sandpile and related models. Physica A 263:4ADSCrossRefGoogle Scholar
  61. Dunn AG, Essam JW, Ritchie DS (1975) Series expansion study of the pair connectedness in bond percolation models. J Phys C 8:4219ADSCrossRefGoogle Scholar
  62. Duplantier B, Saleur H (1989) Exact fractal dimension of 2D Ising clusters. Phys Rev Lett 63:2536ADSMathSciNetCrossRefGoogle Scholar
  63. Essam JW (1980) Percolation theory. Rep Prog Phys 43:833ADSMathSciNetCrossRefGoogle Scholar
  64. Fisher ME (1967a) The theory of condensation and the critical point. Phys NY 3:255Google Scholar
  65. Fisher ME (1967b) Magnetic critical point exponents—their interrelations and meaning. J Appl Phys 38:981ADSCrossRefGoogle Scholar
  66. Fisher ME (1971) The theory of critical point singularities. In: Green MS (ed) Critical phenomena. Proceeding of the international school of physics “Enrico Fermi” course LI, Varenna on lake Como (Italy). Academic, New York, p 1Google Scholar
  67. Fisher ME, Widom B (1969) Decay of correlations in linear systems. J Chem Phys 50:3756ADSCrossRefGoogle Scholar
  68. Flory PJ (1941) Molecular size distribution in three dimensional polymers. I. Gelation. J Am Chem Soc 63:3083CrossRefGoogle Scholar
  69. Flory PJ (1979) Principles of polymer chemistry. Cornell University Press, IthacaGoogle Scholar
  70. Fortuin CM, Kasteleyn PW (1972) On the random-cluster model: I. Introduction and relation to other models. Physica 57:536ADSMathSciNetCrossRefGoogle Scholar
  71. Fortunato S, Satz H (2000) Percolation and deconfinement in SU(2) gauge theory. Nucl Phys B Proc Suppl 83:452ADSMathSciNetMATHCrossRefGoogle Scholar
  72. Fortunato S, Aharony A, Coniglio C, Stauffer D (2004) Number of spanning clusters at the high-dimensional percolation thresholds. Phys Rev E 70:056116ADSCrossRefGoogle Scholar
  73. Frenkel J (1939a) Statistical theory of condensation phenomena. J Chem Phys 7:200ADSCrossRefGoogle Scholar
  74. Frenkel J (1939b) A general theory of heterophase fluctuations and pretransition phenomena. J Chem Phys 7:538ADSCrossRefGoogle Scholar
  75. Gefen Y, Aharony A, Mandelbrot BB, Kirkpatrick S (1981) Solvable fractal family, and its possible relation to the backbone at percolation. Phys Rev Lett 47:1771ADSMathSciNetCrossRefGoogle Scholar
  76. Gimel JC, Nicolai T, Durand D (2001) Monte-Carlo simulation of transient gel formation and break-up during reversible aggregation. Eur Phys J E 5:415CrossRefGoogle Scholar
  77. Given JA, Stell G (1991) Approximations of mean spherical type for lattice percolation models. J Phys A Math Gen 24:3369ADSCrossRefGoogle Scholar
  78. Grest GS, Webman I, Safran SA, Bug ALR (1986) Dynamic percolation in microemulsions. Phys Rev A 33:2842ADSCrossRefGoogle Scholar
  79. Harris AB, Lubensky TC, Holcomb W, Dasgupta C (1975) Renormalization-group approach to percolation problems. Phys Rev Lett 35:327ADSCrossRefGoogle Scholar
  80. Havlin S, Bunde A (1991) Percolation II. In: Bunde A, Havlin S (eds) Fractals and disordered systems. Springer, New York, pp 97–149CrossRefGoogle Scholar
  81. Heermann DW, Stauffer D (1981) Phase diagram for three-dimensional correlated site-bond percolation. Z Phys B 44:339ADSCrossRefGoogle Scholar
  82. Heermann DW, Coniglio A, Klein W, Stauffer D (1984) Nucleation and metastability in three-dimensional Ising models. J Stat Phys 36:447ADSCrossRefGoogle Scholar
  83. Hill TL (1955) Molecular clusters in imperfect gases. J Chem Phys 23:617ADSCrossRefGoogle Scholar
  84. Hu CK (1984) Percolation, clusters, and phase transitions in spin models. Phys Rev B 29:5103ADSCrossRefGoogle Scholar
  85. Hu CK (1992) Histogram Monte Carlo renormalization group method for phase transition models without critical slowing down. Phys Rev Lett 69:2739ADSMathSciNetMATHCrossRefGoogle Scholar
  86. Hu CK, Lin CY (1996) Universal scaling functions for numbers of percolating clusters on planar lattices. Phys Rev Lett 77:8ADSCrossRefGoogle Scholar
  87. Hu CK, Mak KS (1989) Monte Carlo study of the Potts model on the square and the simple cubic lattices. Phys Rev B 40:5007ADSCrossRefGoogle Scholar
  88. Jan N, Coniglio A, Stauffer D (1982) Study of droplets for correlated site-bond percolation in two dimensions. J Phys A 15:L699ADSCrossRefGoogle Scholar
  89. Janke W, Schakel AMJ (2004) Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model. Nucl Phys B 700:385ADSMathSciNetMATHCrossRefGoogle Scholar
  90. Kasteleyn PW, Fortuin CM (1969) Phase transitions in lattice systems with random local properties. J Phys Soc Japan Suppl 26:11ADSGoogle Scholar
  91. Kertesz J (1989) Existence of weak singularities when going around the liquid-gas critical point. Physica A 161:58ADSCrossRefGoogle Scholar
  92. Kertesz J, Coniglio A, Stauffer D (1983) Clusters for random and interacting percolation. In: Deutscher G, Zallen R, Adler J (eds) Percolation structures and processes, vol 5, Annals of the Israel Physical Society. Adam Hilger, Bristol, pp 121–147. The Israel Physical Society, JerusalemGoogle Scholar
  93. Kirkpatrick S (1978) The geometry of the percolation threshold. AIP Conf Proc 40:99ADSCrossRefGoogle Scholar
  94. Klein W, Gould H, Tobochnik J, Alexander FJ, Anghel M, Johnson G (2000) Clusters and fluctuations at mean-field critical points and spinodals. Phys Rev Lett 85:1270ADSCrossRefGoogle Scholar
  95. Ma YG (1999) Application of information theory in nuclear liquid gas phase transition. Phys Rev Lett 83:3617ADSCrossRefGoogle Scholar
  96. Ma YG, Han DD, Shen WQ, Cai XZ, Chen JG, He ZJ, Long JL, Ma GL, Wang K, Wei YB, Yu LP, Zhang HY, Zhong C, Zhou XF, Zhu ZY (2004) Statistical nature of cluster emission in nuclear liquid–vapour phase coexistence. J Phys G Nucl Part Phys 30:13ADSCrossRefGoogle Scholar
  97. Machta J, Newman CM, Stein DL (2007) The Percolation signature of the spin glass transition. J Stat Phys 130:113ADSMathSciNetMATHCrossRefGoogle Scholar
  98. Mader CM, Chappars A, Elliott JB, Moretto LG, Phair L, Wozniak GJ (2003) The three-dimensional Ising model and its Fisher analysis: A paradigm of liquid-vapor coexistence in nuclear multifragmentation. Phys Rev C 68:064601ADSCrossRefGoogle Scholar
  99. Makse HA, Havlin S, Stanley HE (1995) Modelling urban growth patterns. Nature 377:608ADSCrossRefGoogle Scholar
  100. Makse HA, Havlin S, Schwartz M, Stanley HE (1996) Method for generating long-range correlations for large systems. Phys Rev E 53:5445ADSMATHCrossRefGoogle Scholar
  101. Makse HA, Andrade JS Jr, Batty M, Havlin S, Stanley HE (1998) Modeling urban growth patterns with correlated percolation. Phys Rev E 58:7054ADSCrossRefGoogle Scholar
  102. Mallamace F, Chen SH, Liu Y, Lobry L, Micali N (1999) Percolation and viscoelasticity of triblock copolymer micellar solutions. Physica A 266:123ADSCrossRefGoogle Scholar
  103. Mallamace F, Gambadauro P, Micali N, Tartaglia P, Liao C, Chen SH (2000) Kinetic glass transition in a micellar system with short-range attractive interaction. Phys Rev Lett 84:5431ADSCrossRefGoogle Scholar
  104. Mallamace F, Chen SH, Coniglio A, de Arcangelis L, del Gado E, Fierro A (2006) Complex viscosity behavior and cluster formation in attractive colloidal systems. Phys Rev E 73:020402ADSCrossRefGoogle Scholar
  105. Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San FranciscoMATHGoogle Scholar
  106. Martin JE, Adolf D, Wilcoxon JP (1988) Viscoelasticity of near-critical gels. Phys Rev Lett 61:2620ADSCrossRefGoogle Scholar
  107. Mayer JE (1937) The statistical mechanics of condensing systems. I. J Chem Phys 5:67ADSMATHCrossRefGoogle Scholar
  108. Mayer JE, Ackermann PG (1937) The statistical mechanics of condensing systems. II. J Chem Phys 5:74ADSMATHCrossRefGoogle Scholar
  109. Mayer JE, Harrison SF (1938) Statistical mechanics of condensing systems. III. J Chem Phys 6:87ADSCrossRefGoogle Scholar
  110. Mayer JE, Mayer MG (1940) Statistical mechanics. Wiley, New YorkMATHGoogle Scholar
  111. Muller-Krhumbaar H (1974) Percolation in a lattice system with particle interaction. Phys Lett A 50:27ADSCrossRefGoogle Scholar
  112. Murata KK (1979) Hamiltonian formulation of site percolation in a lattice gas. J Phys A 12:81ADSCrossRefGoogle Scholar
  113. Nienhuis B, Berker AN, Riedel EK, Shick M (1979) First- and second-order phase transitions in potts models: Renormalization-group Solution. Phys Rev Lett 43:737ADSCrossRefGoogle Scholar
  114. Odagaki T, Ogita N, Matsuda H (1975) Percolation approach to the metal-insulator transition in super-critical fluid metals. J Phys Soc Jpn 39:618ADSCrossRefGoogle Scholar
  115. Padoa Scioppa C, Sciortino F, Tartaglia P (1998) Coniglio-Klein mapping in the metastable region. Phys Rev E 57:3797ADSCrossRefGoogle Scholar
  116. Pike R, Stanley HE (1981) Order propagation near the percolation threshold. J Phys A 14:L169ADSCrossRefGoogle Scholar
  117. Qian X, Deng Y, Blote HWJ (2005) Dilute Potts model in two dimensions. Phys Rev E 72:056132ADSCrossRefGoogle Scholar
  118. Romano F, Tartaglia P, Sciortino F (2007) Gas–liquid phase coexistence in a tetrahedral patchy particle model. J Phys Condens Matter 19:322101CrossRefGoogle Scholar
  119. Roussenq J, Coniglio A, Stauffer D (1982) Study of droplets for correlated site-bond percolation in three dimensions. J Phys Paris 43:L703CrossRefGoogle Scholar
  120. Safran SA, Webman I, Grest GS (1985) Percolation in interacting colloids. Phys Rev A 32:506ADSCrossRefGoogle Scholar
  121. Sahimi M, Mukhopadhyay S (1996) Scaling properties of a percolation model with long-range correlations. Phys Rev E 54:3870ADSCrossRefGoogle Scholar
  122. Sahimi M, Knackstedt MA, Sheppard AP (2000) Scaling properties of a percolation model with long-range correlations. Phys Rev E 61:4920ADSCrossRefGoogle Scholar
  123. Saika-Voivod I, Zaccarelli E, Sciortino F, Buldyrev SV, Tartaglia P (2004) Effect of bond lifetime on the dynamics of a short-range attractive colloidal system. Phys Rev E 70:041401ADSCrossRefGoogle Scholar
  124. Saleur H, Duplantier B (1987) Exact determination of the percolation hull exponent in two dimensions. Phys Rev Lett 58:2325ADSMathSciNetCrossRefGoogle Scholar
  125. Sator N (2003) Clusters in simple fluids. Phys Rep 376:1ADSMathSciNetCrossRefGoogle Scholar
  126. Skal AS, Shklovskii BI (1975) Topology of an infinite cluster in theory of percolation and its relationship to theory of hopping conduction. Sov Phys Semicond 8:1029Google Scholar
  127. Stanley HE (1977) Cluster shapes at the percolation threshold: And effective cluster dimensionality and its connection with critical-point exponents. J Phys A 10:1211CrossRefGoogle Scholar
  128. Stauffer D (1976) Gelation in concentrated critically branched polymer solutions. Percolation scaling theory of intramolecular bond cycles. J Chem Soc Faraday Trans 72:1354CrossRefGoogle Scholar
  129. Stauffer D (1981) Monte-Carlo simulation of Ising droplets in correlated site-bond percolation. J Phys Lett 42:99CrossRefGoogle Scholar
  130. Stauffer D (1990) Droplets in Ising models. Physica A 168:614ADSMathSciNetCrossRefGoogle Scholar
  131. Stauffer D (1997) Minireview: New results for old percolation. Physica A 242:1. for a minireview on the multiplicity of the infinite clustersADSMathSciNetCrossRefGoogle Scholar
  132. Stauffer D, Aharony A (1994) Introduction to percolation theory. Taylor and Francis, LondonMATHGoogle Scholar
  133. Stauffer D, Coniglio A, Adam M (1982) Gelation and critical phenomena. Polymer Networks 44:103. For a review on percolation and gelation (special volume Polymer networks, Dusek K (ed))CrossRefGoogle Scholar
  134. Stella AL, Vanderzande C (1989) Scaling and fractal dimension of Ising clusters at the d=2 critical point. Phys Rev Lett 62:1067ADSMathSciNetCrossRefGoogle Scholar
  135. Suzuki M (1974) New universality of critical exponents. Progr Theor Phys Kyoto 51:1992ADSCrossRefGoogle Scholar
  136. Swendsen RH, Wang JS (1987) Nonuniversal critical dynamics in Monte Carlo simulations. Phys Rev Lett 58:86ADSCrossRefGoogle Scholar
  137. Sykes MF, Gaunt DS (1976) A note on the mean size of clusters in the Ising model. J Phys A 9:2131ADSCrossRefGoogle Scholar
  138. Tanaka T, Swislow G, Ohmine I (1979) Phase separation and gelation in gelatin gels. Phys Rev Lett 42:1556ADSCrossRefGoogle Scholar
  139. Temesvari T (1984) Multicritical behaviour in the q-state Potts lattice-gas. J Phys A Math Gen 17:1703ADSMathSciNetCrossRefGoogle Scholar
  140. Vernon DC, Plischke M, Joos B (2001) Viscoelasticity near the gel point: A molecular dynamics study. Phys Rev E 64:031505ADSCrossRefGoogle Scholar
  141. Wang JS (1989) Clusters in the three-dimensional Ising model with a magnetic field. Physica A 161:249ADSCrossRefGoogle Scholar
  142. Wang JS, Swendsen R (1990) Cluster Monte Carlo algorithms. Physica A 167:565ADSMathSciNetCrossRefGoogle Scholar
  143. Weinrib A (1984) Long-range correlated percolation. Phys Rev B 29:387ADSMathSciNetCrossRefGoogle Scholar
  144. Weinrib A, Halperin BI (1983) Critical phenomena in systems with long-range-correlated quenched disorder. Phys Rev B 27:413ADSCrossRefGoogle Scholar
  145. Wolff U (1988) Lattice field theory as a percolation process. Phys Rev Lett 60:1461ADSMathSciNetCrossRefGoogle Scholar
  146. Wolff U (1989a) Comparison between cluster Monte Carlo algorithms in the Ising model. Phys Lett B 228:379ADSCrossRefGoogle Scholar
  147. Wolff U (1989b) Collective Monte Carlo updating for spin systems. Phys Rev Lett 62:361ADSCrossRefGoogle Scholar
  148. Wu F (1982) The Potts model. Rev Mod Phys 54:235ADSCrossRefGoogle Scholar
  149. Zaccarelli E (2007) Colloidal gels: Equilibrium and non-equilibrium routes. J Phys Condens Matter 19:323101CrossRefGoogle Scholar

Books and Reviews

  1. Grimmett G (1989) Percolation. Springer, BerlinMATHGoogle Scholar
  2. Sahimi M (1994) Application of percolation theory. Taylor and Francis, LondonGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di Fisica “Ettore Pancini”Università di Napoli “Federico II”, Complesso Universitario di Monte Sant’AngeloNaplesItaly
  2. 2.Istituto SPINCNRNaplesItaly