vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2025-61-1-17-22vestifm-822Research ArticleМАТЕМАТИКАMATHEMATICSПриближенное аналитическое выражение для топологической перколяционной константыApproximate analytical expression for the topological percolation constantГринчукП. С.GrinchukP. S.

Гринчук Павел Семенович – член-корреспондент Национальной академии наук Беларуси, доктор физико-математических наук, заведующий отделением теплофизики

ул. П. Бровки, 15, 220072, Минск

Pavel S. Grinchuk – Corresponding Member of the Na tional Academy of Sciences of Belarus, Dr. Sc. (Physics and Mathematics), Head of the Thermophysics department

15, P. Brovka Str., 220072, Minsk

gps@hmti.ac.byИнститут тепло- и массообмена имени А. В. Лыкова Национальной академии наук БеларусиA. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus2025260320256111722Copyright © Гринчук П.С., 20252025Гринчук П.С.Grinchuk P.S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/822

Получено приближенное аналитическое выражение для топологической перколяционной константы, характеризующей наиболее общие топологические свойства фракталов, прежде всего такие, как связность вблизи порога перколяции.

In this paper, we obtain an approximate analytical expression for the topological percolation constant, which characterizes the most general topological properties of fractals, primarily such as connectivity near the percolation threshold. 

теория перколяциифракталгипотеза Александера – Орбахаpercolation theoryfractalAlexander – Orbach conjectureРабота выполнена в рамках Государственной программы научных исследований «Энергетические и ядерные процессы и технологии», подпрограмма «Энергетические процессы и технологии» (задание 2.31).The work was carried out in the framework of the State Scientific Research Program “Energy and Nuclear Processes and Technologies”, subprogram “Energy Processes and Technologies” (task 2.31).ReferencesStauffer, D. Introduction to Percolation Theory / Stauffer D., Aharony A. – New York: Taylor & Francis, 2018. – 192 p. https://doi.org/10.1201/9781315274386Stauffer D., Aharony A. Introduction to Percolation Theory. New York, Taylor & Francis, 2018. 192 p. https://doi.org/10.1201/9781315274386Sahimi, M. Applications of Percolation Theory / M. Sahimi. – London: CRC Press, 1994. – 276 p. https://doi. org/10.1201/9781482272444Sahimi M. Applications of Percolation Theory. London, CRC Press, 1994. 276 p. https://doi.org/10.1201/9781482272444Grinchuk, P. S. Cluster size distribution in percolation theory and fractal Cantor dust / P. S. Grinchuk // Physical Review E. – 2007. – Vol. 75, № 4. – Art. ID 041118. https://doi.org/10.1103/physreve.75.041118Grinchuk P. S. Cluster size distribution in percolation theory and fractal Cantor dust. Physical Review E, 2007, vol. 75, no. 4, art. ID 041118. https://doi.org/10.1103/physreve.75.041118Grinchuk, P. S. Surfaces of percolation systems in lattice problems / P. S. Grinchuk, O. S. Rabinovich // Physical Review E. – 2003. – Vol. 67, № 4. – Art. ID 046103. https://doi.org/10.1103/physreve.67.046103Grinchuk P. S., Rabinovich O. S. Surfaces of percolation systems in lattice problems. Physical Review E, 2003. vol. 67, no. 4, art. ID 046103. https://doi.org/10.1103/PhysRevE.67.046103Grossman, T. Accessible external perimeters of percolation clusters / T. Grossman, A. Aharony // Journal of Physics A: Mathematical and General. – 1987. – Vol. 20, № 17. – P. L1193. https://doi.org/10.1088/0305-4470/20/17/011Grossman T., Aharony A. Accessible external perimeters of percolation clusters. Journal of Physics A: Mathematical and General, 1987, vol. 20, no. 17, pp. L1193. https://doi.org/10.1088/0305-4470/20/17/011Dynamics of invasion percolation / L. Furuberg, J. Feder, A. Aharony, T. Jossang // Physical Review Letters. – 1988. – Vol. 61, № 18. – Art. ID 2117. https://doi.org/10.1103/physrevlett.61.2117Furuberg L., Feder J., Aharony A., Jossang T. Dynamics of invasion percolation. Physical Review Letters, 1988, vol. 61, no. 18, art. ID 2117. https://doi.org/10.1103/physrevlett.61.2117Sahimi, M. Percolation and polymer morphology and rheology / M. Sahimi // Complex Media and Percolation Theory. Encyclopedia of Complexity and Systems Science Series / eds.: M. Sahimi, A. G. Hunt. – New York: Springer, 2021. https://doi.org/10.1007/978-1-0716-1457-0_388Sahimi M. Percolation and Polymer Morphology and Rheology. Sahimi M., Hunt A. G. (eds.). Complex Media and Percolation Theory. Encyclopedia of Complexity and Systems Science Series. New York, Springer, 2021. https://doi.org/10.1007/978-1-0716-1457-0_388Milovanov, A. V. Fracton pairing mechanism for unconventional superconductors: Self-assembling organic polymers and copper-oxide compounds / A. V. Milovanov, J. J. Rasmussen // Physical Review B. – 2002. – Vol. 66, № 13. – Art. ID 134505. https://doi.org/10.1103/physrevb.66.134505Milovanov A. V., Rasmussen J. J. Fracton pairing mechanism for unconventional superconductors: Self-assembling organic polymers and copper-oxide compounds. Physical Review B, 2002. vol. 66, no. 13, art. ID 134505. https://doi.org/10.1103/physrevb.66.134505Зосимов, В. В. Фракталы в волновых процессах / В. В. Зосимов, Л. М. Лямшев // Успехи физических наук. – 1995. – Т. 165, № 4. – С. 361–402. https://doi.org/10.3367/ufnr.0165.199504a.0361Zosimov V. V., Lyamshev L. M. Fractals in wave processes. Physics-Uspekhi, 1995, vol. 38, no. 4, pp. 347–384. https:// doi.org/10.1070/pu1995v038n04abeh000080Lawler, G. F. The Dimension of the Planar Brownian Frontier is 4/3 / G. F. Lawler, O. Schramm, W. Werner // Mathematical Research Letters. – 2001. – Vol. 8, №. 4. – P. 401–411. https://doi.org/10.4310/mrl.2001.v8.n4.a1Lawler G. F., Schramm O., Werner W. The Dimension of the Planar Brownian Frontier is 4/3. Mathematical Research Letters, 2001, vol. 8, no. 4, pp. 401–411. https://doi.org/10.4310/mrl.2001.v8.n4.a1Зелёный, Л. М. Фрактальная топология и странная кинетика: от теории перколяции к проблемам космической электродинамики / Л. М. Зелёный, А. В. Милованов // Успехи физических наук. – 2004. – Т. 174, № 8. – С. 809– 853. https://doi.org/10.3367/ufnr.0174.200408a.0809Zelenyi L. M., Milovanov A. V. Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics. Physics-Uspekhi, 2004, vol. 47, no. 8, pp. 749–788. https://doi.org/10.1070/pu2004v047n08abeh001705Grinchuk, P. Fractal power law and polymer-like behavior for the metro growth in megacities / P. Grinchuk, S. Danilova-Tretiak // Chaos, Solitons & Fractals. – 2025. – Vol. 194. – Art. ID 116137. https://doi.org/10.1016/j.chaos.2025.116137Grinchuk P., Danilova-Tretiak S. Fractal power law and polymer-like behavior for the metro growth in megacities. Chaos, Solitons & Fractals, 2025, vol. 194, art. ID 116137. https://doi.org/10.1016/j.chaos.2025.116137Alexander, S. Density of states on fractals: «fractons» / S. Alexander, R. Orbach // Journal de Physique Lettres. – 1982. – Vol. 43, № 17. – P. 625–631. https://doi.org/10.1051/jphyslet:019820043017062500Alexander S., Orbach R. Density of states on fractals: «fractons». Journal de Physique Lettres, 1982, vol. 43, no. 17, pp. 625–631. https://doi.org/10.1051/jphyslet:019820043017062500Orbach, R. Dynamics of fractal networks / R. Orbach // Science. – 1986. – Vol. 231, № 4740. – P. 814–819. https://doi.org/10.1126/science.231.4740.814Orbach R. Dynamics of fractal networks. Science, 1986, vol. 231, no. 4740, pp. 814–819. https://doi.org/10.1126/science.231.4740.814Nakayama, T. Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations / T. Nakayama, K. Yakubo, R. L. Orbach // Reviews of Modern Physics. – 1994. – Vol. 66, № 2. – P. 381. https://doi.org/10.1103/revmodphys.66.381Nakayama T., Yakubo K., Orbach R. L. Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Reviews of Modern Physics, 1994, vol. 66, no. 2, pp. 381. https://doi.org/10.1103/revmodphys.66.381Milovanov, A. V. Topological proof for the Alexander-Orbach conjecture / A. V. Milovanov. – Physical Review E. – 1997. – Vol. 56, № 3. – Art. ID 2437. https://doi.org/10.1103/PhysRevE.56.2437Milovanov A. V. Topological proof for the Alexander-Orbach conjecture. Physical Review E, 1997, vol. 56, no. 3, art. ID 2437. https://doi.org/10.1103/PhysRevE.56.2437Фоменко, А. Т. Курс гомотопической топологии / А. Т. Фоменко, Д. Б. Фукс. – М.: Наука, 1989. – 528 с.Fomenko A. T., Fuks D. B. Homotopical Topology. Moscow, Nauka Publ., 1989. 528 p. (in Russian).Milovanov, A. V. Percolation in sign-symmetric random fields: Topological aspects and numerical modeling / A. V. Milovanov, G. Zimbardo // Physical Review E. – 2000. – Vol. 62, № 1. – P. 250. https://doi.org/10.1103/physreve.62.250Milovanov A. V., Zimbardo G. Percolation in sign-symmetric random fields: Topological aspects and numerical modeling. Physical Review E, 2000, vol. 62, no. 1, pp. 250. https://doi.org/10.1103/physreve.62.250Milovanov, A. V. Critical conducting networks in disordered solids: ac universality from topological arguments / A. V. Milovanov, J. J. Rasmussen // Physical Review B. – 2001. – Vol. 64, № 21. – P. 212203. https://doi.org/10.1103/physrevb.64.212203Milovanov A. V., Rasmussen J. J. Critical conducting networks in disordered solids: ac universality from topological arguments. Physical Review B, 2001, vol. 64, no. 21, pp. 212203. https://doi.org/10.1103/physrevb.64.212203

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