Прикрепленные градиентные меры модели SOS, связанные с периодическими граничными законами H_A на деревьях Кэли

Эта работа исследует прикрепленные градиентные меры для моделей SOS (Solid-On-Solid), ассоциированные с H_A-граничными законами периода два, класс которых включает все градиентные Гиббсовы меры периода два, соответствующие пространственно однородному граничному закону. В то время как предыдущие исследования преимущественно фокусировались на пространственно однородных граничных законах и соответствующих ГГМ на деревьях Кэли, данное исследование расширяет анализ, предлагая полную характеристику таких мер. В частности, оно демонстрирует существование прикрепленных градиентных мер на множестве G-допустимых конфигураций и точно оценивает их количество при определенных температурных условиях.

1. Friedli S., Velenik Y. Statistical mechanics of lattice systems. A concrete mathematical introduction. Cambridge University Press, Cambridge, 2018.

2. Georgii H.O. Gibbs Measures and Phase Transitions, Second edition. de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin, 2011.

3. Mukhamedov F. Extremality of Disordered Phase of λ-Model on Cayley Trees. Algorithms, 2022, 15(1), 18 pages.

4. Rozikov U.A. Gibbs measures on Cayley trees, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, pp. xviii+385, 2013.

5. Souissi A., Hamdi T., Mukhamedov F, Andolsi A. On the structure of quantum Markov chains on Cayley trees associated with open quantum random walks, Axioms, 2023, 12(9), 864 pages.

6. Zachary S. Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice. Stochastic Process. Appl., 1985, 20(2), P. 247–256.

7. Funaki T., Spohn H. Motion by mean curvature from the Ginzburg-Landau ∇ϕ interface model. Comm. Math. Phys., 1997, 185.

8. Funaki T. Stochastic interface models. Lectures on Probability Theory and Statistics, Lecture Notes in Math., Springer-Verlag, Berlin, 2015, P. 103–274.

9. Cotar C., Kulske C., Existence of Random Gradient States. ¨ Ann. Appl. Probab., 2012, 22(4), P. 1650–1692.

10. Cotar C., Kulske C., Uniqueness of gradient Gibbs measures with disorder. ¨ Probab. Theory Relat. Fields, 2015, 162, P. 587–635.

11. Van Enter A.C., Kulske C. Non-existence of random gradient Gibbs measures in continuous interface models in ¨ d = 2. Ann. Appl. Probab., 2008, 18, P. 109–119.

12. Henning F., Kulske C. Existence of gradient Gibbs measures on regular trees which are not translation invariant. ¨ Ann. Appl. Probab, 2023, 33(4), P. 3010–3038.

13. Zachary S. Countable state space Markov random fields and Markov chains on trees. Ann. Probab., 1983, 11(4), P. 894–903.

14. Ganikhodjaev N.N., Khatamov N.M., Rozikov U.A. Gradient Gibbs measures of an SOS model with alternating magnetism on Cayley trees. Reports on Mathematical Physics, 2023, 92(3), P. 309–322.

15. Haydarov F.H., Ilyasova R.A., Gradient Gibbs measures with periodic boundary laws of a generalised SOS model on a Cayley tree. J. Stat. Mech. Theory Exp., 2023, vol. 123101, 12 pages.

16. Henning F. Gibbs measures and gradient Gibbs measures on regular trees. PhD Thesis, Ruhr University, 2021, 109 pp.

17. Haydarov F.H., Rozikov U.A. Gradient Gibbs measures of an SOS model on Cayley trees: 4-periodic boundary laws. Reports on Mathematical Physics, 2022, 90(1), P. 81–101.

18. Henning F., Kulske C., Le Ny A., and Rozikov U.A. Gradient Gibbs measures for the SOS model with countable values on a Cayley tree. ¨ Electron. J. Probab, 2019, 24(104), 23 pages.

19. Ilyasova R.A. Height-periodic gradient Gibbs measures for generalised SOS model on Cayley tree. Uzbek Mathematical Journal, 2024, 68(2), P. 92–99.

20. Rozikov U.A. Mirror Symmetry of Height-Periodic Gradient Gibbs Measures of an SOS Model on Cayley Trees. J. Stat. Phys., 2022, 188, 26 pages.

21. Henning F., Kulske C. Height-offset variables and pinning at infinity for gradient Gibbs measures on trees.

22. Kulske C., Schriever P. Gradient Gibbs measures and fuzzy transformations on trees. ¨ Markov Process. Relat. Fields, 2017, 23(4), P. 553–590.

23. Henning F., Kulske C. Coexistence of localized Gibbs measures and delocalized gradient Gibbe measures on trees. ¨ Ann. Appl. Probab, 2021, 31(5), P. 2284–2310.

24. Dobrushin R.L. Description of a random field by means of conditional probabilities and the conditions governing its regularity. Theory Prob. and its Appl., 1968, 13, P. 197–224.

25. Lanford O.E., Ruelle D. Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Physics, 1968, 13, P. 194–215.

26. Ruelle D. Superstable interactions in statistical mechanics. Comm. Math. Physics, 1970, 18, P. 127–159.

27. Khakimov R.M., Haydarov F.H. Translation-invariant and periodic Gibbs measures for the Potts model on the Cayley tree. Theoretical and mathematical physics, 2016, 289, P. 286–295.

28. Prasolov V.V. Polynomials. Springer, Berlin. 2004.

29. Sheffield S. Random surfaces, Large deviations principles and gradient Gibbs measure classifications. PhD Thesis, Stanford University, 2003, 205 pages.