Abstract
We consider inhomogeneous non-oriented Bernoulli bond percolation on \(\mathbb {Z}^d\), where each edge has a parameter depending on its direction. We prove that, under certain conditions, if the sum of the parameters is strictly greater than 1/2, we have percolation in sufficiently high dimensions. The main tool is a dynamical coupling between models for different dimensions with different sets of parameters.
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References
Broadbent, SR., Hammersley, JM.: Percolation processes: I. Crystals and mazes. Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 53, No. 3, 629–641, (1957)
Couto, R.G., de Lima, B.N.B., Sanchis, R.: Anisotropic percolation on slabs. Markov Process. Related Fields 20(1), 145–154 (2014)
Gomes, P..A., Pereira, A., Sanchis, R.: Anisotropic oriented percolation in high dimensions. ALEA Lat. Am. J. Probab. Math. Stat. 17, 531–543 (2020)
Gomes, P.A., Sanchis, R., Silva, R.W.C.: A note on the dimensional crossover critical exponent. Lett. Math. Phys. 110, 3427–3434 (2020)
Gordon, D.M.: Percolation in high dimensions. J. London Math. Soc. 2–44(2), 373–384 (1991)
Grimmett, G.R., Stacey, A.M.: Critical probabilities for site and bond percolation models. Ann. Probab. 26, 1788–1812 (1998)
Grimmett, G.R.: Percolation. Springer-Verlag (1999)
Hara, T., Slade, G.: The self-avoiding-walk and percolation critical points in high dimensions. Comb. Probab. Comput. 4(3), 197–215 (1995)
Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys. 74, 41–59 (1980)
Kesten, H.: Percolation Theory for Mathematicians. Birkhäuser (1982)
Kesten, H.: Asymptotics in high dimensions for percolation. Disorder in physical systems, Oxford University Press, pp 219–240 (1990)
Acknowledgements
The authors thank an anonymous referee who helped improve the readability of the paper and also thank Roger Silva for valuable comments on the first version of the manuscript.
Funding
P.A. Gomes has been supported by São Paulo Research Foundation (FAPESP), grant 2020/02636-3 and grant 2017/10555-0. R. Sanchis has been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), CAPES and by FAPEMIG (APQ-00868-21 and RED-00133-21).
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Gomes, P.A., Pereira, A. & Sanchis, R. Anisotropic Non-oriented Bond Percolation in High Dimensions. J Theor Probab 37, 121–132 (2024). https://doi.org/10.1007/s10959-023-01254-9
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DOI: https://doi.org/10.1007/s10959-023-01254-9