Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-12T19:46:20.350Z Has data issue: false hasContentIssue false

Markov random fields and percolation on general graphs

Published online by Cambridge University Press:  01 July 2016

Olle Häggström*
Affiliation:
Chalmers University of Technology
*
Postal address: Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden.

Abstract

Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the Widom-Rowlinson model and the beach model. Some, but not all, of these implications hold without the bounded degree assumption. We finally give two examples of (random) unbounded degree graphs in which phase transition in all five models can be established: supercritical Galton-Watson trees, and Poisson-Voronoi tessellations of ℝd for d ≥ 2.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aizenman, M., Chayes, J. T., Chayes, L. and Newman, C. M. (1988). Discontinuity of the magnetization in one-dimensional 1/|x-y|2 Ising and Potts models. J. Statist. Phys. 50, 140.CrossRefGoogle Scholar
[2] Benjamini, I. and Schramm, O. (1996). Percolation beyond Zd, many questions and a few answers. Electr. Commun. Prob. 1, 7182.Google Scholar
[3] Benjamini, I. and Schramm, O. (1998). Conformal invariance of Voronoi percolation. Commun. Math. Phys. 197, 75107.CrossRefGoogle Scholar
[4] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Critical percolation on any nonamenable group has no infinite clusters. Ann. Prob., 27, 13471356.Google Scholar
[5] van den Berg, J. (1993). A uniqueness condition for Gibbs measures, with application to the 2–dimensional Ising antiferromagnet. Commun. Math. Phys. 152, 161166.CrossRefGoogle Scholar
[6] van den Berg, J. and Maes, C. (1994). Disagreement percolation in the study of Markov fields. Ann. Prob. 22, 749763.Google Scholar
[7] van den Berg, J. and Steif, J. (1994). Percolation and the hard core lattice gas model. Stoch. Proc. Appl. 49, 179197.Google Scholar
[8] Brightwell, G. R., Häggström, O. and Winkler, P. (1999). Nonmonotonic behavior in hard-core and Widom–Rowlinson models. J. Statist. Phys. 94, 415435.CrossRefGoogle Scholar
[9] Burton, R. and Steif, J. (1994). Nonuniqueness of measures of maximal entropy for subshifts of finite type. Ergod. Th. Dynam. Sys. 14, 213236.Google Scholar
[10] Burton, R. and Steif, J. (1995). New results on measures of maximal entropy. Israel J. Math. 89, 275300.CrossRefGoogle Scholar
[11] Cassi, D. (1992). Phase transition and random walks on graphs: a generalization of the Marmin–Wagner theorem to disordered lattices, fractals, and other discrete structures. Phys. Rev. Lett. 68, 36313634.Google Scholar
[12] Chayes, L. (1996). Percolation and ferromagnetism on Z2: the q-state Potts cases. Stoch. Proc. Appl. 65, 209216.Google Scholar
[13] Chayes, J. T., Chayes, L. and Kotecký, R. (1995). The analysis of the Widom–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551569.Google Scholar
[14] Chayes, L., Kotecký, R. and Shlosman, S. B. (1997). Staggered phases in diluted systems with continuous spins. Commun. Math. Phys. 189, 631640.Google Scholar
[15] Coniglio, A., Nappi, C. R., Peruggi, F. and Russo, L. (1976). Percolation and phase transitions in the Ising model. Commun. Math. Phys. 51, 315323.CrossRefGoogle Scholar
[16] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. 57, 536564.Google Scholar
[17] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, New York.Google Scholar
[18] Georgii, H.-O., Häggström, O. and Maes, C. (1999). The random geometry of equilibrium phases. In Phase transitions and critical phenomena, eds Domb, C. and Lebowitz, J. L.. Academic Press, London, to appear.Google Scholar
[19] Grimmett, G. R. (1989). Percolation. Springer, New York.Google Scholar
[20] Grimmett, G. R. (1994). Percolative problems. In Probability and Phase Transition, ed. Grimmett, G. R.. Kluwer, Dordrecht, pp. 6986.CrossRefGoogle Scholar
[21] Häggström, O., (1996). On phase transitions for subshifts of finite type. Israel J. Math. 94, 319352.Google Scholar
[22] Häggström, O., (1997). Ergodicity of the hard core model on Z2 with parity-dependent activities. Ark. Mat. 35, 171184.Google Scholar
[23] Häggström, O., (1998). Random-cluster representations in the study of phase transitions. Markov Proc. Rel. Fields 4, 275321.Google Scholar
[24] Häggström, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Rand. Struct. Algorithms 9, 295315.Google Scholar
[25] Häggström, O. and Mossel, E. (1998). Nearest-neighbor walks with low predictability profile and percolation in 2+eps dimensions. Ann. Prob. 26, 12121231.CrossRefGoogle Scholar
[26] Häggström, O., Peres, Y. and Schonmann, R. H. (1999). Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness. In Perplexing Probability Problems: Papers in Honor of Harry Kesten, eds Bramson, M. and Durrett, R.. Birkhäuser, Boston, pp. 6990.Google Scholar
[27] Hammersley, J. M. (1957). Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist. 28, 790795.Google Scholar
[28] Hammersley, J. M. (1961). Comparison of atom and bond percolation. J. Math. Phys. 2, 728733.Google Scholar
[29] Holley, R. (1974). Remarks on the FKG inequalities. Commun. Math. Phys. 36, 227231.Google Scholar
[30] Jonasson, J. and Steif, J. (1998). Amenability and phase transition in the Ising model. J. Theor. Prob. 12, 549559.Google Scholar
[31] Kindermann, R. and Snell, J.L. (1980). Markov Random Fields and their Applications. American Mathematical Society, Providence, RI.Google Scholar
[32] Lebowitz, J. L. and Gallavotti, G. (1971). Phase transitions in binary lattice gases. J. Math. Phys. 12, 11291133.Google Scholar
[33] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
[34] Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 7195.Google Scholar
[35] Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Commun. Math. Phys. 125, 337353.Google Scholar
[36] Lyons, R. (1990). Random walks and percolation on trees. Ann. Prob. 18, 931958.CrossRefGoogle Scholar
[37] Lyons, R. and Peres, Y. (1998). Probability on Trees and Networks. In preparation, draft available at http://php.indiana.edu/ rdlyons.Google Scholar
[38] Møller, J., (1994). Lectures on Random Voronoi Tessellations. Springer, New York.Google Scholar
[39] Pemantle, R. and Steif, J. (1999). Robust phase transition for spherical and other models on general trees. Ann. Prob. 27, 876912.Google Scholar
[40] Peres, Y. (1999). Probability on trees: an introductory climb. In Lectures on Probability Theory and Statistics: École d'été de probabilités de Saint-Flour 1997 (Lecture notes in Math. 1717). Springer, Berlin.Google Scholar
[41] Runnels, L. K. and Lebowitz, J. L. (1974). Phase transitions of a multicomponent Widom–Rowlinson model. J. Math. Phys. 15, 17121717.Google Scholar
[42] Schonmann, R. H. and Tanaka, N. (1998). Lack of monotonicity in ferromagnetic Ising model phase diagrams. Ann. Appl. Prob. 8, 234245.Google Scholar
[43] Wheeler, J. C. and Widom, B. (1970). Phase equilibrium and critical behavior in a two-component Bethe-lattice gas or three-component Bethe-lattice solution. J. Chem. Phys. 52, 53345343.CrossRefGoogle Scholar
[44] Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid-vapor phase transition. J. Chem. Phys. 52, 16701684.Google Scholar