Abstract
We give an elementary introduction to the subject of correlated percolation. We illustrate it by diagrams and qualitative data for the Ising model on the square torus of order 100, which serves as a simple model for a binary alloy.
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8. References
J.E. Besag (1974) “Spatial interaction and the statistical analysis of lattice systems.” J.Roy.Stat.Soc. B 36, 192–236.
K. Binder (1976) “Monte Carlo investigations of phase transitions and critical phenomena.” Phase transitions and critical phenomena 5 A, 1–105 (ed. C. Domb and M.S. Green) Academic Press.
A. Coniglio and L. Russo (1979) “Cluster size and shape in random and correlated percolation.” J.Phys., A 12, 545–550.
F. Delyon (1980) “Taille, forme et nombre des amas dans les problèmes de percolation.” Thèse, Université Pierre et Marie Curie, Paris.
A.G. Dunn, J.W. Essam and J.M. Lovelock (1975) “Scaling theory for the pair-connectedness in percolation models.” J.Phys. C 8, 743–750.
A.G. Dunn, J.W. Essam and D.S. Ritchie (1975) “Series expansion study of the pair connectedness in bond percolation models.” J.Phys. C 8 4219–4235.
A.G. Dunn, J.W. Essam and D.S. Ritchie (1975) “Series expansion study of the pair connectedness in bond percolation models.” J.Phys. C 8, 4219–4235.
R. Durrett (1981) “An introduction to infinite particle systems.” Stochastic Processes and Applic. 11, 109–150.
J.R. Ehrman, L.D. Fosdick and D.C. Handscomb (1960) “Computation of order parameters in an Ising lattice by the Monte Carlo Method.” J.Math.Phys. 1 547–558.
J.W. Essam (1972) “Percolation and cluster size.” Phase transitions and critical phenomena 2, 197–270 (ed. C.Domb and M.S. Green) Academic Press.
M.E. Fisher (1961) “Critical probabilities for cluster size and percolation problems.” J.Math.Phys. 2, 620–627.
A. Flammang (1977) “Percolation cluster sizes and perimeters in three dimensions.” Z.Phys. B 28, 47–50.
C.M. Fortuin and P.W. Kasteleyn (1972) “On the random cluster model. I. Introduction and relation to other models.” Physica 57, 536–564.
C.M. Fortuin (1972) “On the random cluster model. II. The percolation model.” Physica 58 393–418.
C.M. Fortuin (1972) “On the random cluster model. III. The simple random cluster model.” Physica 59, 545–570.
L.D. Fosdick (1959) “Calculation of order parameters in a binary alloy by the Monte Carlo method.” Phys.Rev. 116, 565–573.
J.W. Halley (1983) “Polychromatic percolation.” Percolation structure and processes (ed. G. Deutscher, R. Zallen and J. Adler) Ann.Israel Phys.Soc. 5, 323–352. Adam Hilger, Bristol.
J.M. Hammersley (1957) “Percolation processes. II. The connective constant.” Proc.Camb.Phil.Soc. 53, 642–645.
J.M.Hammersley and D.C. Handscomb (1964) Monte Carlo methods. Methuen.
J.M.Hammersley (1975) “Rumination on infinite Markov systems.” Perspectives in probability and statistics, 195–200.
H. Kesten (1982) Percolation theory for mathematicians. Birkhäuser, Boston.
R.Kindermann and J.L.Snell (1982) Markov random fields and their applications. Contemporary Mathematics, 1. (Amer.Math.Soc.).
H. Kunz and B. Souillard (1978) “Essential singularity and asymptotic behaviour of cluster size distribution.” J.Stat.Phys. 19, 77–106.
D.P. Landau (1977) “Critical behaviour of a bcc Ising antiferromagnet in a magnetic field”. Phys.Rev. B 16, 4164–4170.
P.L. Leath (1976) “Cluster shape and critical exponents near percolation threshold.” Phys.Rev.Lett. 36, 921–924.
P.L. Leath (1976) “Cluster size and boundary distribution near percolation threshold.” Phys.Rev. B 14, 5046–5055.
P.L. Leath and G.R. Reich (1978) “Ramification of large clusters near the percolation threshold.” J.Phys. C 11, 1155–1168.
N. Metropolis, A.W. Rosenbluth, N.M. Rosenbluth, A.H. Teller and T.E. Teller (1953) “Equations of state calculations by fast computing machine.” J.Chem.Phys. 21, 1087–1092.
J. Moussouris (1974) “Gibbs and Markov random systems with constraints.” J.Stat.Phys. 10, 11–33.
E. Müller-Hartmann and J. Zittartz (1977) “Interface free energy and transition temperature of the square lattice Ising antiferromagnet at finite magnetic field.” Z.Phys. B 27, 261–266.
G.F. Newell and E.W. Montroll (1953) “Theory of the Ising model of ferromagnetism.” Rev.Mod.Phys. 25, 353–389.
C.M. Newman and L.S. Schulman (1981) “Infinite clusters in percolation models.” J.Stat.Phys. 26, 613–628.
L. Onsager (1944) “Crystal statistics I. A two-dimensional model with an order-disorder transition.” Phys.Rev. 65, 117–149.
L. Onsager (1949) “The spontaneous magnetization of the Ising model.” Suppl.Nuovo Cimento 6, 241–243.
D.C. Rapaport (1978) “Monte Carlo study of the phase boundary of the Ising antiferromagnet.” Phys.Lett. 65 A, 147–148.
V.K.S. Shante and S. Kirkpatrick (1971) “An introduction to percolation theory.” Adv.Phys. 20, 325–357.
I. Sneddon (1979) “Ising antiferromagnets in a magnetic field.” J.Phys. C 12, 3051–3057.
D. Stauffer (1977) “Exact distribution of cluster size.” Z.Phys. B 25, 391–399.
D. Stauffer (1978) “Scaling assumption for lattice animals in percolation theory.” J.Stat.Phys. 18, 125–136.
E. Stoll and C. Domb (1979) “Shape and size of two-dimensional percolation clusters with and without correlations.” J.Phys. A 12 1843–1855.
M.F. Sykes and J.W. Essam (1964) “Critical percolation probabilities by series methods.” Phys.Rev. A 133, 310–315.
E.T.Whittaker and G.N.Watson (1940) A course of modern analysis. (4th ed.), Cambridge University Press.
A.P. Young and R.B. Stinchcombe (1975) “A renormalization group theory for percolation problems.” J.Phys. C 8, L535–L540
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Hammersley, J.M., Mazzarino, G. (1983). Markov fields, correlated percolation, and the Ising model. In: Hughes, B.D., Ninham, B.W. (eds) The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation. Lecture Notes in Mathematics, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073261
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DOI: https://doi.org/10.1007/BFb0073261
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