Powstawanie kropli dla wielokątnych pól Markowa

• Autorzy: Schreiber Tomasz

Identyfikatory

DOI

10.14708/wm.v44i01.5081

• Języki publikacji: PL

Abstrakty

EN

The paper discusses a class of continuum random contour models in the plane, going under the name of polygonal Markov fields as originally introduced by Arak & Surgailis and sharing a number of crucial features with the two-dimensional Ising model, which makes them interesting from the viewpoint of mathematical statistical physics. For such systems, modeling the co-existence of two opposing phases separated by polygonal contours, we present our results on low-temperature geometry of phase-separating interfaces. In this context, we show that in the phase transition regime the surplus of dominated phase creates a disk-shaped droplet surrounded by ocean of dominating phase (Wulff body) and minimising the model-specific surface energy functional. The proof is based on a particular graphical construction which also found its applications in digital image segmentation as indicated at the end of this article.

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Wiadomości Matematyczne
• Rocznik: 2008
• Tom: T. 44, Nr 1
• Strony: 15--22
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Schreiber Tomasz

• Wydział Matematyki i Informatyki UMK ul. Chopina 12/18, 87-100 Toruń

Bibliografia

• [1] T. Arak and D. Surgailis, Markov Fields with Polygonal Realisations, Probab. Th. Rel. Fields, 80 (1989), 543-579.
• [2] T. Arak and D. Surgailis, Consistent polygonal fields, Probab. Th. Rel. Fields, 89 (1991), 319-346.
• [3] P. Arak, P. Clifford and D. Surgailis, Point-based polygonal models for random graphs, Adv. Appl. Probab., 25 (1993), 348-372.
• [4] R. Dobrushin, R. Kotecký and S. Shlosman, Wulff construction - a global shape from local interaction, Translations of Mathematical Monographs, AMS, 104 (1992), American Mathematical Society, Providence.
• [5] H.-O. Georgii, О. Häggström and C. Maes, The random geometry of equilibrium phases, in: C. Dоmb and J. L. Lebowitz (eds.), Phase transitions and critical phenomena, 18 (2000), 1-142, Academic Press, London.
• [6] R. Kluszczyński, M. N. M. van Lieshout and T. Schreiber, An algorithm for binary image segmentation using polygonal Markov fields, w F. Roli, S. Vitulanо (Ed.), Image Analysis and Processing, Proceedings of the 13th International Conference on Image Analysis and Processing, Lecture Notes in Computer Science, 3615 (2005), 383-390.
• [7] R. Kluszczyński, M. N. M. van Lieshout and T. Schreiber, Image segmentation by polygonal Markov fields, przyjęta do druku w Annals of the Institute of Mathematical Statistics (2007),
• [8] D. Ioffe and R. Schonmann, Dobrushin-Kotecký-Shlosman theory up to the critical temperature, Comm. Math. Phys., 199 (1998), 117-167.
• [9] L. Onsager, Crystal statistics. I. A two-dimensional model with order-disorder phase transition, Phys. Rev., 2 (1944), 117-149.
• [10] T. Schreiber, Random dynamics and thermodynamic limits for polygonal Markov fields in the plane, Adv. Appl. Probab., 37 (2005), 884-907.
• [11] T. Schreiber, Dobrushin-Kotecký-Shlosman theorem for polygonal Markov fields in the plane, Journal of Statistical Physics, 123 (2006), 631-684.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).