Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-08T00:25:15.913Z Has data issue: false hasContentIssue false
  • English
  • Français

Random dynamics and thermodynamic limits for polygonal Markov fields in the plane

Part of: Equilibrium statistical mechanics Geometric probability and stochastic geometry Special processes

Published online by Cambridge University Press:  01 July 2016

Tomasz Schreiber
Tomasz Schreiber*
Affiliation:
Nicolaus Copernicus University
*
∗ Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, 87-100, Poland. Email address: tomeks@mat.uni.torun.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernández et al. (1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

Keywords

Polygonal Markov fieldrandom dynamicsMetropolis simulationperfect simulationthermodynamic limitphase transition

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B21: Continuum models (systems of particles, etc.)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 884 - 907
Copyright
Copyright © Applied Probability Trust 2005 

References

Arak, T. (1983). On Markovian random fields with finite numbers of values. In Proc. 4th USSR–Japan Symp. Prob. Theory Math. Statist. (Tbilisi, USSR, 1982), eds Ito, K. and Prokhorov, Yu. V., Springer, New York.Google Scholar
Arak, T. and Surgailis, D. (1989). Markov fields with polygonal realisations. Prob. Theory Relat. Fields 80, 543–579.Google Scholar
Arak, T. and Surgailis, D. (1991). Consistent polygonal fields. Prob. Theory Relat. Fields 89, 319–346.Google Scholar
Arak, T., Clifford, P. and Surgailis, D. (1993). Point-based polygonal models for random graphs. Adv. Appl. Prob. 25, 348–372.Google Scholar
Clifford, P. and Nicholls, G. (1994). A Metropolis sampler for polygonal image reconstruction. Preprint, available at http://www.stats.ox.ac.uk/∼clifford/papers/met_poly.html.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Fernández, R., Ferrari, P. and Garcia, N. (1998). Measures on contour, polymer or animal models. A probabilistic approach. Markov Process. Relat. Fields 4, 479–497.Google Scholar
Fernández, R., Ferrari, P. and Garcia, N. (2002). Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102, 63–88.Google Scholar
Liggett, T. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston, MA.Google Scholar
Nicholls, G. K. (2001). Spontaneous magnetisation in the plane. J. Statist. Phys. 102, 1229–1251.Google Scholar
Schreiber, T. (2004). Mixing properties for polygonal Markov fields in the plane. Submitted.Google Scholar
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.Google Scholar
Surgailis, D. (1991). Thermodynamic limit of polygonal models. Acta Appl. Math. 22, 77–102.Google Scholar