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

  • Tomasz Schreiber (a1)

Abstract

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.

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Copyright

Corresponding author

Postal address: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, 87-100, Poland. Email address: tomeks@mat.uni.torun.pl

References

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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.
Arak, T. and Surgailis, D. (1989). Markov fields with polygonal realisations. Prob. Theory Relat. Fields 80, 543579.
Arak, T. and Surgailis, D. (1991). Consistent polygonal fields. Prob. Theory Relat. Fields 89, 319346.
Arak, T., Clifford, P. and Surgailis, D. (1993). Point-based polygonal models for random graphs. Adv. Appl. Prob. 25, 348372.
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.
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Fernández, R., Ferrari, P. and Garcia, N. (1998). Measures on contour, polymer or animal models. A probabilistic approach. Markov Process. Relat. Fields 4, 479497.
Fernández, R., Ferrari, P. and Garcia, N. (2002). Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102, 6388.
Liggett, T. (1985). Interacting Particle Systems. Springer, New York.
Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston, MA.
Nicholls, G. K. (2001). Spontaneous magnetisation in the plane. J. Statist. Phys. 102, 12291251.
Schreiber, T. (2004). Mixing properties for polygonal Markov fields in the plane. Submitted.
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.
Surgailis, D. (1991). Thermodynamic limit of polygonal models. Acta Appl. Math. 22, 77102.

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

  • Tomasz Schreiber (a1)

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