Approximate analytical expression for the topological percolation constant
https://doi.org/10.29235/1561-2430-2025-61-1-17-22
Abstract
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.
Supporting agencies: 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).
About the Author
P. S. Grinchuk
A. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus
Belarus
Belarus
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
References
1. Stauffer D., Aharony A. Introduction to Percolation Theory. New York, Taylor & Francis, 2018. 192 p. https://doi.org/10.1201/9781315274386
2. Sahimi M. Applications of Percolation Theory. London, CRC Press, 1994. 276 p. https://doi.org/10.1201/9781482272444
3. Grinchuk 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.75.041118
4. Grinchuk 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.67.046103
5. Grossman 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/10.1088/0305-4470/20/17/011
6. Furuberg 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.61.2117
7. Sahimi 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_388
8. Milovanov 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.66.134505
9. Zosimov V. V., Lyamshev L. M. Fractals in wave processes. Physics-Uspekhi, 1995, vol. 38, no. 4, pp. 347–384. https:// doi.org/10.1070/pu1995v038n04abeh000080
10. Lawler 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.a1
11. Zelenyi 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/pu2004v047n08abeh001705
12. Grinchuk 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.2025.116137
13. Alexander 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:019820043017062500
14. Orbach R. Dynamics of fractal networks. Science, 1986, vol. 231, no. 4740, pp. 814–819. https://doi.org/10.1126/science.231.4740.814
15. Nakayama 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.66.381
16. Milovanov 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.56.2437
17. Fomenko A. T., Fuks D. B. Homotopical Topology. Moscow, Nauka Publ., 1989. 528 p. (in Russian).
18. Milovanov 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.62.250
19. Milovanov 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
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