Abstract
We study the 2D-Ising model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FK-percolation events that concern the phenomenon of phase coexistence.
References
Alexander K.S.: On weak mixing in lattice models. Probab. Theory Relat. Fields 110, 441–471 (1998)
Alexander K.S.: Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32, 441–487 (2004)
Alexander K.S.: Stability of the Wulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation. Probab. Theory Relat. Fields 91, 507–532 (1992)
Alexander K.S.: Cube-root boundary fluctuations for droplets in random cluster models. Comm. Math. Phys. 224, 733–781 (2001)
Alexander K.S., Chayes J.T., Chayes L.: The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Comm. Math. Phys. 131, 1–50 (1990)
Barbato, D.: Tesi di Laurea. Universita di Pisa (2001)
Bodineau T.: The Wulff construction in three and more dimensions. Comm. Math. Phys. 207, 197–229 (1999)
Bodineau T.: Slab percolation for the Ising model. Probab. Theory Relat. Fields 132, 83–118 (2005)
Bricmont J., Lebowitz J.L., Pfister C.E.: On the local structure of the phase separation line in the two-dimensional Ising system. J. Stat. Phys. 26, 313–332 (1981)
Camia, F., Newman, C.M.: The Full Scaling Limit of Two-Dimensional Critical Percolation. Preprint (2005)
Cerf, R.: Large deviations for three-dimensional supercritical percolation. Astérisque 267 (2000)
Cerf, R.: The Wulff crystal in Ising and percolation models. Ecole d’été de probabilités, Saint Flour (2004)
Cerf, R., Messikh, R.J.: On the Wulf crystal of the 2D-Ising model near criticality (2006)
Cerf R., Pisztora Á.: On the Wulff crystal in the Ising model. Ann. Probab. 28, 947–1017 (2000)
Cerf R., Pisztora Á.: Phase coexistence in Ising, Potts and percolation models. Ann. I. H. P. PR 37, 643–724 (2001)
Chayes J.T., Chayes L., Schonmann R.H.: Exponential decay of connectivities in the two-dimensional Ising model. J. Stat. Phys. 49, 433–445 (1987)
Couronné O., Messikh R.J.: Surface order large deviations for 2D FK-percolation and Potts models. Stoch. Proc. Appl. 113, 81–99 (2004)
Dobrushin, R.L., Kotecký, R., Shlosman, S.B.: Wulff construction: a global shape from local interaction. Am. Math. Soc. Transl. Ser. (1992)
Edwards R.G., Sokal A.D.: Generalization of the Fortuin–Kasteleyn–Swenden–Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988)
Ellis R.S.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)
Fortuin C.M., Kasteleyn R.W.: On the random-cluster model I. Introduction and relation to other models. Physica 57, 125–145 (1972)
Grimmett, G.R.: Percolation and disordered systems. Lectures on Probability Theory and Statistics. In: Bertrand, P. (ed.) Lectures from the 26th Summer School on Probability Theory held in Saint Flour, August 19–September 4, 1996. Lecture Notes in Mathematics 1665 (1997)
Grimmett, G.R.: The random cluster model. In: Kelly, F.P. (ed.) Probability, Statistics and Optimization: A Tribute to Peter Whittle, pp. 49–63 (1994)
Grimmett G.R.: The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23, 1461–1510 (1995)
Grimmett G.R., Marstrand J.M.: The supercritical phase of percolation is well behaved. Proc. R. Soc. Lond. Ser. A 430, 439–457 (1990)
Hoeffding W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963)
Ioffe D.: Large deviation for the 2D Ising model: a lower bound without cluster expansions. J. Stat. Phys. 74, 411–432 (1993)
Ioffe D.: Exact large deviation bounds up to T c for the Ising model in two dimensions. Probab. Theory Relat. Fields 102, 313–330 (1995)
Ioffe D., Schonmann R.: Dobrushin–Kotecký–Shlosman theorem up to the critical temperature. Comm. Math. Phys. 199, 117–167 (1998)
Laanait L., Messager A., Ruiz J.: Phase coexistence and surface tensions for the Potts model. Comm. Math. Phys. 105, 527–545 (1986)
Mc Coy B.M., Wu T.T.: The Two Dimensional Ising Model. Harvard University Press, Cambridge (1973)
Messikh, R.J.: The surface tension of the 2D Ising model near criticality (2006, submitted)
Messikh, R.J.: From the Ising model towards the Black and White Mumford–Shah functional. Ph.D. thesis (2004)
Onsager L.: Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. 65, 117–149 (1944)
Pfister C.E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64, 953–1054 (1991)
Pfister C.E., Velenik Y.: Large deviations and continuum limit in the 2D Ising model. Probab. Theory Relat. Fields 109, 435–506 (1997)
Pisztora Á.: Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Relat. Fields 104, 427–466 (1996)
Smirnov S.: Critical percolation in the plane: conformal invariance. Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris 333, 239–244 (2001)
Slade, G.: The Lace Expansion and its Applications. Ecole d’été de probabilités, Saint Flour (2004)
Smirnov S., Werner W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8, 729–744 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cerf, R., Messikh, R.J. The 2D-Ising model near criticality: a FK-percolation analysis. Probab. Theory Relat. Fields 150, 193–217 (2011). https://doi.org/10.1007/s00440-010-0272-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0272-0
Keywords
- Large deviations
- Criticality
- Phase coexistence
Mathematics Subject Classification (2000)
- 60F10