Abstract
In this note we consider long-range q-states Potts models on Z d, d≥ 2. For various families of non-summable ferromagnetic pair potentials φ(x)≥ 0, we show that there exists, for all inverse temperature β > 0, an integer N such that the truncated model, in which all interactions between spins at distance larger than N are suppressed, has at least q distinct infinite-volume Gibbs states. This holds, in particular, for all potentials whose asymptotic behaviour is of the type φ(x)∼ ‖x‖−α, 0≤α≤ d. These results are obtained using simple percolation arguments.
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M. Aizenman, J. T. Chayes, L. Chayes, and C. N. Newman. Discontinuity of the magnetization in one-dimensional \(1/|x-y|^2\) Ising and Potts models. Journ. Stat. Phys., 50:1–40, 1988.
M. Aizenman, H. Kesten, and C. N. Newman. Uniqueness of the infinite cluster and continuity of the connectivity functions for short and long range percolation. Commun. Math. Phys., 111:505–531, 1987.
N. Berger. Transience, recurrence and critical behavior for long-range percolation. Commun. Math. Phys., 226:531–558, 2002.
C. E. Bezuidenhout, G. R. Grimmett, and H. Kesten. Strict inequality for critical values of Potts models and random-cluster processes. Commun. Math. Phys., 158:1–16, 1993.
P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.
M. Biskup, L. Chayes, and N. Crawford. Mean-field driven first-order phase transitions in systems with long-range interactions. To appear in Journ. Stat. Phys., 2005.
T. Bodineau. Slab percolation for the Ising model. Probab. Theory and Rel. Fields, 132(1):83–118, 2005.
A. Bovier and M. Zahradník. The low-temperature phases of Kac-Ising models. Journ. Stat. Phys., 87:311–332, 1997.
S. A. Cannas, A. de Magalhães, and F. A. Tamarit. Evidence of exactness of the mean field theory in the nonextensive regime of long-range spin models. Phys. Rev. Lett., 122:597–607, 1989.
S. A. Cannas and F. A. Tamarit. Long-range interactions and nonextensivity in ferromagnetic spin models. Phys. Rev. B, 54:R12661–R12664, 1986.
M. Cassandro and E. Presutti. Phase transitions in Ising systems with long but finite range interactions. Mark. Proc. Rel. Fields, 2:241–262, 1996.
F. J. Dyson. Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys., 12:91–107, 1969.
C. Fortuin. On the random cluster model. II. Physica, 58:393–418, 1972.
C. Fortuin. On the random cluster model. III. Physica, 59:545–570, 1972.
C. Fortuin and P. Kasteleyn. On the random cluster model. I. Physica, 57:536–564, 1972.
S. Friedli, B. N. B. de Lima, and V. Sidoravicius. On long range percolation with heavy tails. Elect. Comm. in Probab., 9:175–177, 2004.
S. Friedli and C.-É. Pfister. Non-analyticity and the van der Waals limit. Journ. Stat. Phys., 114(3/4):665–734, 2004.
J. Fröhlich and T. Spencer. The phase transition in the one-dimensional Ising model with 1/r 2 interaction energy. Commun. Math. Phys., 84:87–101, 1982.
H.-O. Georgii. Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, 1988.
H.-O. Georgii, O. Häggström, and C. Maes. The random geometry of equilibrium phases. In C. Domb and J. Lebowitz, editors, Phase Transitions and Critical Phenomena Vol. 18, pages 1–142. Academic Press, London, 2001.
G. Grimmett, M. Keane, and J. M. Marstrand. On the connectedness of a random graph. Math. Proc. Cambridge Phil. Soc., 96:151–166, 1984.
G. Grimmett and J. M. Marstrand. The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser, A 430:439–457, 1990.
P. Hertel, H. Narnhofer, and W. Thirring. Thermodynamic functions for fermions with gravostatic and electrostatic interactions. Commun. Math. Phys., 28:159–176, 1972.
M. Kac, G. E. Uhlenbeck, and P. C. Hemmer. On the van der Waals theory of the vapor-liquid equilibrium. Journ. Math. Phys., 4:216–228, 1962.
H. Kesten. Asymptotics in high dimensions for percolation. In O. S. Publ., editor, Disorder in physical systems, pages 219–240. Oxford Univ. Press, New York, 1990.
J. L. Lebowitz and O. Penrose. Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition. Journ. Math. Phys., 7:98–113, 1966.
T. M. Liggett, R. H. Schonmann, and A. M. Stacey. Domination by product measures. Ann. Probab., 25:71–95, 1997.
R. Meester and J. E. Steif. On the critical value for long range percolation in the exponential case. Commun. Math. Phys., 180:483–504, 1996.
S. A. Pirogov and Y. G. Sinai. Phase diagrams of classical lattice systems. Teoreticheskaya i Matematicheskaya Fizika, 26(1):61–76, 1976.
V. Sidoravicius, D. Surgailis, and M. E. Vares. On the truncated anisotropic long-range percolation on Z 2. Stoch. Proc. and Appl., 81:337–349, 1999.
F. Tamarit and C. Anteneodo. Rotators with long-range interactions: Connections with the mean-field approximation. Phys. Rev. Lett., 84:208, 2000.
B. P. Vollmayr-Lee and E. Luijten. Kac-potential treatment of nonintegrable interactions. Phys.Rev. E, 63:031108, 2001.
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Work supported by Swiss National Foundation for Science, Conselho Nacional de Desenvolvimento Cientìfico e Tecnològico, and Programa de Auxìlio para Recèm Doutores PRPq-UFMG.
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Friedli, S., de Lima, B.N.B. On the Truncation of Systems with Non-Summable Interactions. J Stat Phys 122, 1215–1236 (2006). https://doi.org/10.1007/s10955-005-8023-9
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DOI: https://doi.org/10.1007/s10955-005-8023-9