Abstract
We consider the so-called length-interacting Arak-Surgailis polygonal Markov fields with V-shaped nodes — a continuum and isometry invariant process in the plane sharing a number of properties with the two-dimensional Ising model. For these polygonal fields we establish a low-temperature phase separation theorem in the spirit of the Dobrushin-Kotecký-Shlosman theory, with the corresponding Wulff shape deteremined to be a disk due to the rotation invariant nature of the considered model. As an important tool replacing the classical cluster expansion techniques and very well suited for our geometric setting we use a graphical construction built on contour birth and death process, following the ideas of Férnandez, Ferrari and Garcia.
Similar content being viewed by others
References
T. Arak, On Markovian random fields with finite number of values, 4th USSR-Japan symposium on probability theory and mathematical statistics, Abstracts of Communications, Tbilisi (1982).
T. Arak and D. Surgailis, Markov Fields with Polygonal Realisations. Probab. Th. Rel. Fields 80:543–579 (1989).
T. Arak and D. Surgailis, Consistent polygonal fields. Probab. Th. Rel. Fields 89:319–346 (1991).
T. Arak, P. Clifford and D. Surgailis, Point-based polygonal models for random graphs. Adv. Appl. Probab. 25:348–372 (1993).
Yu. Baryshnikov and J. E. Yukich, Gaussian Limits for Random Measures in Geometric Probability. Annals of Appl. Prob. 15:213–253 (2005).
T. Bodineau, D. Ioffe and I. Velenik, Rigorous probabilistic analysis of equilibrium crystal shapes. Journal of Statistical Physics 41:1033–1098 (2000).
R. Dobrushin, R. Kotecký and S. Shlosman, Wulff construction: a global shape from local interaction, AMS translation series, Providence RI,104 (1992).
R. Dobrushin and S. Shlosman, Large and moderate deviations in the Ising model. Adv. in Soviet Math. 20:91–220 (1994).
R. Fernández, P. Ferrari and N. Garcia, Measures on contour, polymer or animal models. A probabilistic approach. Markov Processes and Related Fields 4:479–497 (1998).
R. Fernández, P. Ferrari and N. Garcia, Loss network representation of Ising contours. Ann. Probab. 29:902–937 (2001).
R. Fernández, P. Ferrari and N. Garcia, Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Proc. Appl. 102:63–88 (2002).
D. Ioffe and R. Schonmann, Dobrushin-Kotecký-Shlosman theory up to the critical temperature, Comm. Math. Phys. 199:117–167 (1998).
G. K. Nicholls, Spontaneous magnetisation in the plane. Journal of Statistical Physics 102:1229–1251 (2001).
T. Schreiber, Mixing properties for polygonal Markov fields in the plane, submitted, available at: http://www.mat.uni.torun.pl/preprints, 18-2003 (2004a).
T. Schreiber, Random dynamics and thermodynamic limits for polygonal Markov fields in the plane, to appear in Advances in Applied Probability 37.4 (2005), available at: http://www.mat.uni.torun.pl/preprints, 17-2004 (2004b).
D. Surgailis, Thermodynamic limit of polygonal models. Acta applicandae mathematicae 22:77–102 (1991).
S. Wey, Un théorème limite local, C. R. Acad. Sci. Paris; Sér. I, 320:997–1002 (1995).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schreiber, T. Dobrushin-Kotecký-Shlosman Theorem for Polygonal Markov Fields in the Plane. J Stat Phys 123, 631–684 (2006). https://doi.org/10.1007/s10955-006-9053-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9053-7