Geodesic
Hierarchical models in discrete percolation theory and Markov branching processes
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 1, Tome 235 (2024), pp. 15-33.

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A brief introduction to percolation theory is given. Within the framework of the discrete percolation theory on infinite graphs, we develop a method for approximating the percolation probability based on the construction of a sequence of infinite graphs of a special type called the hierarchical graphs. The calculation of the percolation probability for graphs of this type is reduced to the analysis of a suitable Markov branching process with discrete time.
Keywords: infinite graph, percolation probability, branching random process, supercritical regime, connectedness relation 
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Yu. P. Virchenko; D. A. Cherkashin. Hierarchical models in discrete percolation theory and Markov branching processes. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 1, Tome 235 (2024), pp. 15-33. http://geodesic.mathdoc.fr/item/INTO_2024_235_a1/

[1] Antonova E. S., Virchenko Yu. P., “Svoistvo monotonnosti veroyatnosti perkolyatsii bernullievskikh sluchainykh polei na beskonechnykh grafakh”, Nauch. ved. BelGU. Ser. Fiz. Mat., 11 (82):20 (2010), 28–61

[2] Virchenko Yu. P., “Perkolyatsiya”, Matematicheskaya fizika. Entsiklopediya, M., 1998

[3] Virchenko Yu. P., “Periodicheskii graf”, Matematicheskaya fizika. Entsiklopediya, Rossiiskaya entsiklopediya, M., 1998

[4] Virchenko Yu. P., Tolmacheva Yu. A., “Mazhorantnye otsenki poroga perkolyatsii bernullievskogo polya na kvadratnoi reshetke”, Ukr. mat. zh., 57:10 (2005), 1315–1326 | MR | Zbl

[5] Virchenko Yu. P., Shpilinskaya O. L., “Tochechnye sluchainye polya s markovskimi izmelcheniyami i geometriya fraktalno neuporyadochennykh sred”, Teor. mat. fiz., 124:3 (2000), 490–505 | DOI | MR | Zbl

[6] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

[7] Menshikov M. V., Molchanov S. A., Sidorenko A. F., “Teoriya perkolyatsii i nekotorye prilozheniya”, Itogi nauki i tekhniki. Teoriya veroyatnostei, matematicheskaya statistika i teoreticheskaya kibernetika. T. 2, VINITI, M., 1986, 53–110

[8] Molchanov S. A., Stepanov A. K., “Prosachivanie sluchainykh polei, I”, Teor. mat. fiz., 55:2 (1983), 246–256 | MR

[9] Molchanov S. A., Stepanov A. K., “Prosachivanie sluchainykh polei, II”, Teor. mat. fiz., 55:3 (1983), 419–430 | MR

[10] Molchanov S. A., Stepanov A. K., “Prosachivanie sluchainykh polei, III”, Teor. mat. fiz., 65:3 (1985), 371–379

[11] Sevastyanov B. A., Vetvyaschiesya protsessy, Nauka, M., 1971

[12] Sinai Ya. G., Teoriya fazovykh perekhodov. Strogie rezultaty, Nauka, M., 1980

[13] Tarasevich Yu. Yu., Perkolyatsiya: teoriya, prilozheniya, algoritmy, Editorial URSS, M., 2002

[14] Cherkashin D. A., Virchenko Yu. P., “Ierarkhicheskie reshetochnye modeli teorii perkolyatsii”, Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniya–XXXV, ed. Kondaurova D. E., Izd. dom VGU, Voronezh, 2024, 364–366

[15] Antonova E. S., Virchenko Yu. P., “Monotonicity of the probability of percolation for Bernoulli random fields on periodic graphs”, J. Math. Sci., 175:1 (2011), 86–90 | DOI | MR | Zbl

[16] Barucha-Reid A. T., Elements of the Theory of Markov Processes and Their Applications, Mc Grow-Hill, New York, 1960 | MR

[17] Broadbent S. R., Hammersley J. M., “Percolation processes, I. Crystals and mazes”, Proc. Cambridge Phil. Soc., 53 (1957), 629–641 | DOI | MR | Zbl

[18] Essam J. W., “Percolation Theory”, Rep. Prog. Phys., 43 (1986), 833–912 | DOI | MR

[19] Frisch C. M., Hammersley J. M., “Percolation processes and related topics”, J. SIAM., 11 (1963), 894–918 | MR

[20] Grimmett G., Percolation, Springer-Verlag, New York, 1999 | MR | Zbl

[21] Hammersley J. M., “Percolation processes: lower bounds for the critical probability”, Ann. Math. Stat., 28:3 (1957), 790–795 | DOI | MR | Zbl

[22] Kesten H., Percolation Theory for Mathematicians, Birkhäuser, Boston, 1982 | MR | Zbl

[23] Nummelin E., General Irreducible Markov Chains and Non-negative Operators, Cambridge Univ. Press, New York, 1984 | MR | Zbl

[24] Ruelle D., Statistical Mechanics. Rigorous Results, New York, 1969 | MR | Zbl

[25] Simon R., The $P(\varphi)_2$ Euclidian (Quantum) Field Theory, Princeton Univ. Press, Princeton, New Jersey, 1974 | MR

[26] Virchenko Yu. P., Shpilinskaya O. L., “Spaces of random sets in $\mathbb{R}^d$”, Lobachevsky Math. J., 44:3 (2023), 1043–1059 | DOI | MR

[27] Virchenko Yu. P., Tolmacheva Yu. A., “Method of Sequential Approximative Estimates in Discrete Percolation Theory”, Studies in Mathematical Physics Research, ed. Benton C. V., Nova Science, New York, 2004, 155–175 | MR