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References
- D. Aldous. Random walks on finite groups and rapidly mixing Markov chains. Seminar on probability, XVII, 243--297, Lecture Notes in Math., 986, Springer, Berlin, 1983 . Math. Review 86j:60156
- T. Bodineau. Translation invariant Gibbs states for the Ising model. Probab. Theory Related Fields 135 (2006), no. 2, 153--168. Math. Review 2007b:82007
- R. Bubley, M. Dyer. Path Coupling: a technique for proving rapid mixing in Markov Chains. Proc. of the 38th Annual IEEE Symposium on Foundations of Computer Science, 1997 , 223--231.
- N. Berger, C. Kenyon, E. Mossel, Y. Peres. Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Relat. Fields 131 (2005), no. 3, 311--340. Math. Review 2005k:82066
- T. Bodineau, F. Martinelli. Some new results on the kinetic Ising model in a pure phase. J. Statist. Phys. 109 (2002), no. 1-2, 207--235. Math. Review 2003i:82057
- I. Benjamini, O. Schramm. Percolation beyond $\mathbb{Z}^d$, many questions and a few answers. Electron Comm. Probab. 1 (1996), no.8, 71--82. Math. Review 97j:60179
- I. Benjamini, O. Schramm. Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 (2001), no. 2, 487--507. Math. Review 2002h:82049
- J.T. Chayes, L. Chayes, J.P. Sethna, D.J. Thouless. A mean field spin glass with short-range interactions. Commun. Math. Phys. 106 (1986),no. 1, 41--89. Math. Review 88i:82054
- D. Fisher, D. Huse. Dynamics of droplet fluctuations in pure and random Ising systems. Phys. Rev. B 35 (1987), no.13, 6841--6846.
- H.-O. Georgii. Gibbs measures and Phase Transitions. de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 1988 . Math. Review 89k:82010
- O. Hâ°ggstrËm, J. Jonasson, R. Lyons. Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30 (2002), no. 1, 443--473. Math. Review 2003e:60220
- S. Hoory, N. Linial, A. Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439--561. Math. Review 2007h:68055
- D. Ioffe. Extremality of the disordered state for the Ising model on general trees. Trees (Versailles, 1995) , 3--14, Progr. Probab., 40, Birkhâ°user, Basel, 1996. Math. Review 98j:82013
- J. Jonasson. The random cluster model on a general graph and a phase transition characterization of nonamenability. Stochastic Process. Appl. 79 (1999), no. 2, 335--354. Math. Review 99k:60249
- J. Jonasson, J.E. Steif. Amenability and phase transition in the Ising model. J. Theoret. Probab. 12 (1999), no. 2, 549--559. Math. Review 2000b:60238
- H. Kesten. Percolation theory for mathematicians. Progr. Probab. Statist. 2. Birkhâ°user, Boston, Mass., 1982. Math. Review 84i:60145
- C. Kenyon, E. Mossel, Y. Peres. Glauber dynamics on trees and hyperbolic graphs. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), IEEE Computer Soc., Los Alamitos, CA, 2001 , 568--578.
- T. Lindvall. Lectures on the coupling method. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1992. Math. Review 94c:60002
- R. Lyons. The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 (1989), no. 2, 337--353. Math. Review 90h:82046
- R. Lyons. Phase transitions on Nonamenable Graphs. J. Math. Phys. 41 (2000), no.3, 1099--1126. Math. Review 2001c:82028
- M. Luby, D. Randall, A. Sinclair. Markov chain algorithms for planar lattice structures. SIAM J. Comput. 31 (2001), no. 1, 167--192. Math. Review 2002h:82046
- W. Magnus. Noneuclidean tesselations and their groups. Pure and Applied Mathematics, Vol. 61. Academic Press, New York-London, 1974. . Math. Review 50 #4774
- F. Martinelli. Lectures on Glauber dynamics for discrete spin models. Lectures on probability and statistics (Saint-Flour, 1997), 93--191, Lecture Notes in Math., 1717, Springer, Berlin, 1999 . Math. Review 2002a:60163
- F. Martinelli, E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region I: The attractive case. Comm. Math. Phys. 161 (1994), no. 3, 447--486. Math. Review 96c:82040
- F. Martinelli, E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region II: The general case. Comm. Math. Phys. 161 (1994), no. 3, 487--514. Math. Review 96c:82041
- F. Martinelli, A. Sinclair, D. Weitz. Glauber dynamics on trees: boundary conditions and mixing time. Comm. Math. Phys. 250 (2004), no. 2, 301--334. Math. Review 2005j:82052
- R. Rietman, B. Nienhuis, J. Oitmaa. Ising model on hyperlattices. J. Phys. A 25 (1992), no. 24, 6577-6592. Math. Review 94e:82026
- L. Saloff-Coste. Lectures on finite Markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996) , 301-413, Lecture Notes in Math., 1665 , Springer, Berlin, 1997 . Math. Review 99b:60119
- R. H. Schonmann. Multiplicity of Phase Transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 (2001), no. 2, 271--322. Math. Review 2002h:82036
- C. M. Series, Ya. G. Sinai. Ising models on the Lobachevsky plane. Comm. Math. Phys. 128 (1990), no. 1, 63--76. Math. Review 91b:82006
- D.W. Stroock, B. Zegarlinski. The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 (1992), no. 1, 175--194. Math. Review 93j:82013
- D. Weitz. Combinatorial criteria for uniqueness of the Gibbs measure. Random Structures Algorithms 27 (2005), no. 4, 445--475. Math. Review 2006k:82036
- C. C. Wu. Ising models on Hyperbolic Graphs. J. Stat. Phys. 85 (1997), no. 1-2, 251--259. Math. Review 98a:82028
- C. C. Wu. Ising models on Hyperbolic Graphs II. J. Statist. Phys. 100 (2000), no. 5-6, 893--904. Math. Review 2002h:82027

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