najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-2-154-163najo-11Research ArticleMATHEMATICSМАТЕМАТИКАPinned gradient measures of SOS model associated with HA-boundary laws on Cayley treesПрикрепленные градиентные меры модели SOS, связанные с периодическими граничными законами HA на деревьях Кэлиhttps://orcid.org/0000-0001-9388-122XХайдаровФ. Х.HaydarovF. H.

Хайдаров Фарход Халимжонович

Farhod H. Haydarov

9, University str., Tashkent, 100174

54, Mustaqillik Ave., Tashkent, 100007

haydarovimc@mail.ruhttps://orcid.org/0009-0003-2145-2196ИльясоваР. А. қизиIlyasovaR. A.

Ильясова Рисолат Акмал қизи

Risolat A. Ilyasova

54, Mustaqillik Ave., Tashkent, 100007

University str., 4 Olmazor district, Tashkent, 100174

ilyasova.risolat@mail.ruhttps://orcid.org/0000-0001-8511-4856МамаюсуповХ. С. углиMamayusupovK. S.

Мамаюсупов Худойор Султонтош угли

Khudoyor S. Mamayusupov

54, Mustaqillik Ave., Tashkent, 100007

k.mamayusupov@newuu.uzV. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences; New Uzbekistan UniversityUzbekistanNew Uzbekistan University; National University of UzbekistanUzbekistanNew Uzbekistan UniversityUzbekistan202519052025162154163Copyright © Haydarov F.H., Ilyasova R.A., Mamayusupov K.S., 20252025Хайдаров Ф.Х., Ильясова Р.А., Мамаюсупов Х.С.Haydarov F.H., Ilyasova R.A., Mamayusupov K.S.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/11

This paper investigates pinned gradient measures for SOS (Solid-On-Solid) models associated with HA-boundary laws of period two, a class that encompasses all 2-height periodic gradient Gibbs measures corresponding to a spatially homogeneous boundary law. While previous research has predominantly focused on a spatially homogeneous boundary law and corresponding GGMs on Cayley trees, this study extends the analysis by providing a comprehensive characterization of such measures. Specifically, it demonstrates the existence of pinned gradient measures on a set of G-admissible configurations and precisely quantifies their number under certain temperature conditions.

Эта работа исследует прикрепленные градиентные меры для моделей SOS (Solid-On-Solid), ассоциированные с HA-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.

SOS модельградиентная конфигурацияG-допустимая конфигурациязначения спиновдерево Кэлиградиентная мераградиентная мера Гиббсадва периодических граничных законаSOS modelgradient configurationG-admissible configurationspin valuesCayley treegradient measuregradient Gibbs measuretwo periodic boundary lawWe sincerely thank the referee for their valuable and insightful comments, which have helped improve this work.ReferencesFriedli S., Velenik Y. Statistical mechanics of lattice systems. A concrete mathematical introduction. Cambridge University Press, Cambridge, 2018.Friedli S., Velenik Y. Statistical mechanics of lattice systems. A concrete mathematical introduction. Cambridge University Press, Cambridge, 2018.Georgii H.O. Gibbs Measures and Phase Transitions, Second edition. de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin, 2011.Georgii H.O. Gibbs Measures and Phase Transitions, Second edition. de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin, 2011.Mukhamedov F. Extremality of Disordered Phase of λ-Model on Cayley Trees. Algorithms, 2022, 15(1), 18 pages.Mukhamedov F. Extremality of Disordered Phase of λ-Model on Cayley Trees. Algorithms, 2022, 15(1), 18 pages.Rozikov U.A. Gibbs measures on Cayley trees, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, pp. xviii+385, 2013.Rozikov U.A. Gibbs measures on Cayley trees, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, pp. xviii+385, 2013.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.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.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.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.Funaki T., Spohn H. Motion by mean curvature from the Ginzburg-Landau ∇ϕ interface model. Comm. Math. Phys., 1997, 185.Funaki T., Spohn H. Motion by mean curvature from the Ginzburg-Landau ∇ϕ interface model. Comm. Math. Phys., 1997, 185.Funaki T. Stochastic interface models. Lectures on Probability Theory and Statistics, Lecture Notes in Math., Springer-Verlag, Berlin, 2015, P. 103–274.Funaki T. Stochastic interface models. Lectures on Probability Theory and Statistics, Lecture Notes in Math., Springer-Verlag, Berlin, 2015, P. 103–274.Cotar C., Kulske C., Existence of Random Gradient States. ¨ Ann. Appl. Probab., 2012, 22(4), P. 1650–1692.Cotar C., Kulske C., Existence of Random Gradient States. ¨ Ann. Appl. Probab., 2012, 22(4), P. 1650–1692.Cotar C., Kulske C., Uniqueness of gradient Gibbs measures with disorder. ¨ Probab. Theory Relat. Fields, 2015, 162, P. 587–635.Cotar C., Kulske C., Uniqueness of gradient Gibbs measures with disorder. ¨ Probab. Theory Relat. Fields, 2015, 162, P. 587–635.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.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.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.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.Zachary S. Countable state space Markov random fields and Markov chains on trees. Ann. Probab., 1983, 11(4), P. 894–903.Zachary S. Countable state space Markov random fields and Markov chains on trees. Ann. Probab., 1983, 11(4), P. 894–903.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.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.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.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.Henning F. Gibbs measures and gradient Gibbs measures on regular trees. PhD Thesis, Ruhr University, 2021, 109 pp.Henning F. Gibbs measures and gradient Gibbs measures on regular trees. PhD Thesis, Ruhr University, 2021, 109 pp.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.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.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.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.Ilyasova R.A. Height-periodic gradient Gibbs measures for generalised SOS model on Cayley tree. Uzbek Mathematical Journal, 2024, 68(2), P. 92–99.Ilyasova R.A. Height-periodic gradient Gibbs measures for generalised SOS model on Cayley tree. Uzbek Mathematical Journal, 2024, 68(2), P. 92–99.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.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.Henning F., Kulske C. Height-offset variables and pinning at infinity for gradient Gibbs measures on trees.Henning F., Kulske C. Height-offset variables and pinning at infinity for gradient Gibbs measures on trees.Kulske C., Schriever P. Gradient Gibbs measures and fuzzy transformations on trees. ¨ Markov Process. Relat. Fields, 2017, 23(4), P. 553–590.Kulske C., Schriever P. Gradient Gibbs measures and fuzzy transformations on trees. ¨ Markov Process. Relat. Fields, 2017, 23(4), P. 553–590.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.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.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.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.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.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.Ruelle D. Superstable interactions in statistical mechanics. Comm. Math. Physics, 1970, 18, P. 127–159.Ruelle D. Superstable interactions in statistical mechanics. Comm. Math. Physics, 1970, 18, P. 127–159.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.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.Prasolov V.V. Polynomials. Springer, Berlin. 2004.Prasolov V.V. Polynomials. Springer, Berlin. 2004.Sheffield S. Random surfaces, Large deviations principles and gradient Gibbs measure classifications. PhD Thesis, Stanford University, 2003, 205 pages.Sheffield S. Random surfaces, Large deviations principles and gradient Gibbs measure classifications. PhD Thesis, Stanford University, 2003, 205 pages.

The authors declare that there are no conflicts of interest present.