Coloring percolation clusters at random

Stochastic Processes and their Applications - Tập 96 - Trang 213-242 - 2001
Olle Häggström1
1Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden

Tài liệu tham khảo

Aizenman, 1987, The phase boundary in dilute and random Ising and Potts ferromagnets, J. Phys. A, 20, L313, 10.1088/0305-4470/20/5/010 Aizenman, 1988, Discontinuity of the magnetization in one-dimensional 1/|x−y|2 Ising and Potts models, J. Statist. Phys., 50, 1, 10.1007/BF01022985 Benjamini, I., Lyons, R., Peres, Y., Schramm, O., 2001. Uniform spanning forests, Ann. Probab. 29, 1–65. Benjamini, I., Schramm, O., 1999. Recent progress on percolation beyond Zd. http://www.wisdom.weizmann.ac.il/ ̃schramm/papers/pyond-rep/index.html. van den Berg, 1993, A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet, Comm. Math. Phys., 152, 161, 10.1007/BF02097061 Biskup, 2000, Gibbs states of graphical representations of the Potts model with external fields, J. Math. Phys., 41, 1170, 10.1063/1.533183 Campanino, 1985, An upper bound for the critical percolation probability for the three-dimensional cubic lattice, Ann. Probab., 13, 478, 10.1214/aop/1176993004 Chayes, 1998, Graphical representations for Ising systems in external fields, J. Statist. Phys., 93, 17, 10.1023/B:JOSS.0000026726.43558.80 Coniglio, 1976, Percolation and phase transition in the Ising model, Comm. Math. Phys., 51, 315, 10.1007/BF01617925 van Enter, 1996, On the possible failure of the Gibbs property for measures on lattice systems, Markov Process Related Fields, 2, 209 van Enter, 1993, Regularity properties of position-space renormalization group transformations: scope and limitations of Gibbsian theory, J. Statist. Phys., 72, 879, 10.1007/BF01048183 van Enter, 2000, The Griffiths singularity random field, 59 van Enter, 2000, Dobrushin's program on Gibbsianity restoration: weakly Gibbs and almost Gibbs random fields, 51 Gandolfi, 1988, On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation, Ann. Probab., 16, 1147, 10.1214/aop/1176991681 Georgii, 1981, Spontaneous magnetization of randomly dilute ferromagnets, J. Statist. Phys., 25, 369, 10.1007/BF01010795 Georgii, 1988 Georgii, 2001, The random geometry of equilibrium phases, 1, 10.1016/S1062-7901(01)80008-2 Grimmett, 1995, The stochastic random-cluster process, and the uniqueness of random-cluster measures, Ann. Probab., 23, 1461, 10.1214/aop/1176987791 Grimmett, 1999 Häggström, 1995, Random-cluster measures and uniform spanning trees, Stochastic Process Appl., 59, 267, 10.1016/0304-4149(95)00042-6 Häggström, 1996, Almost sure quasilocality fails for the random-cluster model on a tree, J. Statist. Phys., 84, 1351, 10.1007/BF02174134 Häggström, 1996, A note on (non-)monotonicity in temperature for the Ising model, Markov Process Related Fields, 2, 529 Häggström, 1999, Positive correlations in the fuzzy Potts model, Ann. Appl. Probab., 9, 1149, 10.1214/aoap/1029962867 Häggström, 2000, The Ising model on diluted graphs and strong amenability, Ann. Probab., 28, 1111, 10.1214/aop/1019160327 Harris, 1960, A lower bound on the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc., 56, 13, 10.1017/S0305004100034241 Kesten, 1980, The critical probability of bond percolation on the square lattice equals 12, Comm. Math. Phys., 74, 41, 10.1007/BF01197577 Liggett, 1997, Domination by product measures, Ann. Probab., 25, 71, 10.1214/aop/1024404279 Lyons, 1990, Random walks and percolation on trees, Ann. Probab., 18, 931, 10.1214/aop/1176990730 Lyons, 1992, Random walks, capacity and percolation on trees, Ann. Probab., 20, 2043, 10.1214/aop/1176989540 Lyons, 2000, Phase transitions on nonamenable graphs, J. Math. Phys., 41, 1099, 10.1063/1.533179 Lyons, R., Peres, Y., 2001. Probability on Trees and Networks, in preparation. Current version available at http://php.indiana.edu/ ̃rdlyons/. Lyons, 1999, Indistinguishability of percolation clusters, Ann. Probab., 27, 1809, 10.1214/aop/1022677549 Maes, 1999, Almost Gibbsian versus weakly Gibbsian measures, Stochastic Process Appl., 79, 1, 10.1016/S0304-4149(98)00083-0 Newman, 1981, Infinite clusters in percolation models, J. Statist. Phys., 26, 613, 10.1007/BF01011437 Peres, 1998, The number of infinite clusters in dynamical percolation, Probab. Theory Related Fields, 111, 141, 10.1007/s004400050165 Pfister, 1995, Almost sure quasilocality in the random cluster model, J. Statist. Phys., 79, 765, 10.1007/BF02184883 Propp, 1996, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Struct. Algorithms, 9, 223, 10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O Swendsen, 1987, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett., 58, 86, 10.1103/PhysRevLett.58.86