## The Annals of Probability

### The Ising model on diluted graphs and strong amenability

#### Abstract

Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external field.We show that for nonamenable graphs, for Bernoulli percolation with $p$ close to 1, all the infinite clusters have persistent transition.On the other hand, we show that for transitive amenable graphs, the infinite clusters for any stationary percolation do not have persistent transition. This extends a result of Georgii for the cubic lattice. A geometric consequence of this latter fact is that the infinite clusters are strongly amenable (i.e., their anchored Cheeger constant is 0). Finally we show that the critical temperature for the Ising model with no external field on the infinite clusters of Bernoulli percolation with parameter $p$, on an arbitrary bounded degree graph, is a continuous function of $p$.

#### Article information

Source
Ann. Probab., Volume 28, Number 3 (2000), 1111-1137.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160327

Digital Object Identifier
doi:10.1214/aop/1019160327

Mathematical Reviews number (MathSciNet)
MR1797305

Zentralblatt MATH identifier
1023.60085

#### Citation

Häggström, Olle; Schonmann, Roberto H.; Steif, Jeffrey E. The Ising model on diluted graphs and strong amenability. Ann. Probab. 28 (2000), no. 3, 1111--1137. doi:10.1214/aop/1019160327. https://projecteuclid.org/euclid.aop/1019160327

#### References

• [1] Adams, S. (1992). Følner independence and the amenable Ising model. Ergodic Theory Dynam. Systems 12 633-657.
• [2] Aizenman, M., Chayes, J., Chayes, L. and Newman, C. M. (1987). The phase boundary of dilute and random Ising and Potts ferromagnets. J. Phys. A 20 I313-I318.
• [3] Babson, E. and Benjamini, I. (1999). Cut sets and normed cohomology with application to percolation. Proc. Amer. Math. Soc. 127 589-597.
• [4] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29-66.
• [5] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 1347-1356.
• [6] Benjamini, I., Lyons, R. and Schramm, O. (1999). Percolation perturbations in potential theory and random walks. In Random Walks and Discrete Potential Theory (M. Picardello and W. Woess, eds.) 56-84. Cambridge Univ. Press.
• [7] Benjamini, I. and Schramm, O. (1996). Percolation beyond Zd, many questions and a few answers. Electr. Comm. Probab. 1 71-82.
• [8] Benjamini, I. and Schramm, O. (2000). Recent progress on percolation beyond Zd. Available at http://www.wisdom.weizmann.ac.il/ schramm/papers/pyond-rep/index.html.
• [9] Benjamini, I. and Schramm, O. (2000). Percolation in the hyperbolic plane. Preprint.
• [10] Berg, J. van den and Steif, J. (1999). On the existence and nonexistence of finitary codings for a class of random fields. Ann. Probab. 27 1501-1522.
• [11] Bleher, P. M., Ruiz, J. and Zagrebnov, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 473-482.
• [12] Burton, R. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501-505.
• [13] Burton, R. and Keane, M. (1991). Topological and metric properties of infinite clusters in stationary two-dimensional site percolation. Israel J. Math. 76 299-316.
• [14] Chayes, J., Chayes, L. and Fr ¨ohlich, J. (1985). The low-temperature behavior of disordered magnets. Comm. Math. Phys. 100 399-437.
• [15] Chen, D. and Peres, Y. (2000). Anchored expansion, percolation and speed. Preprint.
• [16] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York.
• [17] Evans, W., Kenyon, C., Peres, Y. and Schulman, L. J. (1999). Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 410-433.
• [18] Falk, H. and Gehring, G. A. (1975). Correlation function and transition temperature bounds for bond-disordered Ising systems. J. Phys. C 8 L298-L301.
• [19] Georgii, H.-O. (1981). Spontaneous magnetization of randomly dilute ferromagnets. J. Statist. Phys. 25 369-396.
• [20] Georgii, H.-O. (1984). On the ferromagnetic and the percolative region of random spin systems. Adv. in Appl. Probab. 16 732-765.
• [21] Georgi, H.-O. (1985). Disordered Ising ferromagnets and percolation. In Particle Systems, Random Media and Large Deviations (R. Durrett, ed.) Amer. Math. Soc., Providence, RI. 135-147.
• [22] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, New York.
• [23] Greenleaf, F. P. (1973). Ergodic theorems and the construction of summing sequences in amenable locally compact groups. Comm. Pure Appl. Math. 26 29-46.
• [24] Griffiths, R. B. and Lebowitz, J. L. (1968). Random spin systems: some rigorous results. J. Math. Phys. 9 1284-1292.
• [25] Grimmett, G. R. and Newman, C. M. (1990). Percolation in + 1 dimensions. In Disorder in Physical Systems (G. R. Grimmett, and D. J. A. Welsh, eds.) 219-240. Clarendon Press, Oxford.
• [26] Grimmett, G. R. and Stacey, A. M. (1998). Critical probabilities for site and bond percolation models. Ann. Probab. 26 1788-1812.
• [27] H¨aggstr ¨om, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423-1436.
• [28] H¨aggstr ¨om, O. (1998). Random-cluster representations in the study of phase transitions. Markov Process Related Fields 4 275-321.
• [29] H¨aggstr ¨om, O. (1999). Markov random fields and percolation on general graphs. Adv. in Appl. Probab. 32 39-66.
• [30] H¨aggstr ¨om, O. and Peres, Y. (1999). Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously. Probab. Theory Related Fields 113 273-285.
• [31] H¨aggstr ¨om, 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: Festscrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) 69-90. Birkh¨auser, Boston.
• [32] Holley, R. (1974). Remarks on the FKG inequalities. Comm. Math. Phys. 36 227-231.
• [33] Ioffe, D. (1996). On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37 137-143.
• [34] Ioffe, D. (1996). Extremality of the disordered state for the Ising model on general trees. In Trees (B. Chauvin, S. Cohen and A. Roualt, eds.) 3-14. Birkh¨auser, Basel.
• [35] Jonasson, J. (1999). The random cluster model on a general graph and a phase transition characterization of nonamenability. Stochastic Process Appl. 79 335-534.
• [36] Jonasson, J. and Steif, J. (1999). Amenability and phase transition in the Ising model. J. Theoret. Probab. 12 549-559.
• [37] Kaimanovich, V. and Vershik, A. (1983). Random walks on discrete groups: boundary and entropy. Ann. Probab. 11 457-490.
• [38] Kesten, H. (1959). Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 336-354.
• [39] Kensten, H. (1959). Full Banach mean on countable groups. Math. Scand. 7 146-156.
• [40] Lalley, S. (1998). Percolation on Fuchsian groups. Ann. Inst. H. Poincar´e Probab. Statist 34 151-178.
• [41] Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Probab. 25 71-95.
• [42] Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337-353.
• [43] Lyons, R. (1990). Random walk and percolation on trees. Ann. Probab. 18 931-958.
• [44] Lyons, R. (1992). Random walk, capacity and percolation on trees. Ann. Probab. 20 2043-2088.
• [45] Lyons, R. (1995). Random walks and the growth of groups. C. R. Acad. Sci. Paris Ser. I Math. 320 1361-1366.
• [46] Lyons, R., Pemantle, R. and Peres, Y. (1996). Random walks on the lamplighter group. Ann. Probab. 24 1993-2006.
• [47] Lyons, R. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809-1836.
• [48] Newman, C. M. and Schulman, L. S. (1981). Infinite clusters in percolation models. J. Statist. Phys. 26 613-628.
• [49] Newman, C. M. and Wu, C. C. (1990). Markov fields on branching planes. Probab. Theory Related Fields 85 539-552.
• [50] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223-252.
• [51] Schonmann, R. H. (1999). Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields 113 287-300.
• [52] Schonmann, R. H. (1999). Percolation in + 1 dimensions at the uniqueness threshold. In Perplexing Probability Problems: Festschrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) 53-67. Birkh¨auser, Boston.
• [53] Schonmann, R. H. and Tanaka, N. I. (1998). Lack of monotonicity in ferromagnetic Ising model phase diagrams. Ann. Appl. Probab. 8 234-245.
• [54] Series, C. M. and Sinai, Y. G. (1990). Ising models on the Lobachevsky plane. Comm. Math. Phys. 128 63-76.
• [55] Soardi, P. M. and Woess, W. (1990). Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math.205 471-486.
• [56] Vir´ag, B. (2000). Anchored expansion and random walk. Geom. Funct. Anal. To appear.
• [57] Woess, W. (1991). Topological groups and infinite graphs. Discrete Math. 95 373-384.
• [58] Wu, C. C. (1993). Critical behavior of percolation and Markov fields on branching planes. J. Appl. Probab. 30 538-547.
• [59] Wu, C. C. (1996). Ising models on hyperbolic graphs. J. Statist. Phys. 85 251-259.
• [60] Zhang, Y. (1996). Continuity of percolation probability in +1 dimensions. J. Appl. Probab. 33 427-433.