Abstract
We consider the discrete Boolean model of percolation on weighted graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Assouad, P.: Étude d’une dimension métrique liée à la possibilité de plongements dans \({\bf R}^{n}\). C. R. Acad. Sci. Paris Sér. A-B 288 15, 1379–1401 (1979)
Benjamini, I., Schramm, O.: Recurrence of Distributional Limits of Finite Planar Graphs, pp. 533–545. Springer, New York (2011). https://doi.org/10.1007/978-1-4419-9675-6_15
Breuillard, E.: Geometry of locally compact groups of polynomial growth and shape of large balls. arXiv preprint arXiv:0704.0095 (2007)
Chan, H.T., Gupta, A., Maggs, B.M., Zhou, S.: On hierarchical routing in doubling metrics. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 762–771. Society for Industrial and Applied Mathematics (2005)
Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications. Wiley, Hoboken (2013)
Coniglio, A., Nappi, C.R., Peruggi, F., Russo, L.: Percolation and phase transitions in the ising model. Commun. Math. Phys. 51(3), 315–323 (1976)
Daley, D., Vere-Jones, D.: An Introduction to the Theory of Point Processes, vol. 1, 2nd edn. Springer, New York (2005)
de Bruijn, N.: Algebraic theory of Penrose’s non-periodic tilings of the plane. Indag. Math. (Proc.) 84(1), 39–52 (1981). https://doi.org/10.1016/1385-7258(81)90016-0
Dolye, P.G.: Application of Rayleigh’s short-cut method to Pólya’s recurrence problem. Ph.D. thesis, Dartmouth College (1982)
Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. Mathematical Association of America, Washington DC (1984)
Galves, A., Garcia, N., Löcherbach, E., Orlandi, E., et al.: Kalikow-type decomposition for multicolor infinite range particle systems. Ann. Appl. Probab. 23(4), 1629–1659 (2013)
Georgii, H.O., Häggström, O., Maes, C.: The Random Geometry of Equilibrium Phases; Phase Transitions and Critical Phenomena, vol. 18. Academic Press, London (2001)
Gilbert, E.N.: Random plane networks. J. Soc. Ind. Appl. Math. 9(4), 533–543 (1961)
Gouéré, J.B.: Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36, 1209–1220 (2008)
Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Mathématiques de l’IHÉS 53(1), 53–78 (1981)
Imrich, W., Seifter, N.: A survey on graphs with polynomial growth. Discret. Math. 95(1), 101–117 (1991)
Kendall, W.S.: Notes on perfect simulation. Markov Chain Monte Carlo 7, 93–146 (2005)
Krön, B.: Growth of self-similar graphs. J. Graph Theory 45(3), 224–239 (2004)
LaFontaine, J., Katz, M., Gromov, M., Bates, S.M., Pansu, P., Pansu, P., Semmes, S.: Metric Structures for Riemannian and non-Riemannian Spaces. Springer, New York (2007)
Lehrbäck, J., Tuominen, H.: A note on the dimensions of assouad and aikawa. J. Math. Soc. Jpn. 65(2), 343–356 (2013). https://doi.org/10.2969/jmsj/06520343
Lyons, R., Peres, Y.: Probability on Trees and Networks, vol. 42. Cambridge University Press, Cambridge (2016)
Mackay, J.M., Tyson, J.T.: Conformal Dimension: Theory and Application, vol. 54. American Mathematical Society, Providence (2010)
Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)
Meester, R., Roy, R.: Continuum Percolation. Cambridge University Press, Cambridge (1996)
Pansu, P.: Métriques de carnot-carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math 129(1), 1–60 (1989)
Pechersky, E., Yambartsev, A.: Percolation properties of the non-ideal gas. J. Stat. Phys. 137(3), 501–520 (2009)
Propp, J.G., Wilson, D.B.: Exact sampling with coupled markov chains and applications to statistical mechanics. Random Struct. Algorithms 9(1–2), 223–252 (1996)
Raghunathan, M.: Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, New York (1972)
Seifter, N., Woess, W.: Approximating graphs with polynomial growth. Glasg. Math. J. 42(1), 1–8 (2000)
Stoyan, D., Mecke, K.: The boolean model: from matheron till today. Space, Structure and Randomness, pp. 151–181. Springer, New York (2005)
Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: Proceedings of the Thirty-sixth Annual ACM Symposium on Theory of computing, pp. 281–290. ACM (2004)
Telcs, A.: Volume and time doubling of graphs and random walks: the strongly recurrent case. Commun. Pure Appl. Math. 54(8), 975–1018 (2001)
Telcs, A.: The art of random walks. 1885. Springer, New York (2006)
Wolf, J.A., et al.: Growth of finitely generated solvable groups and curvature of riemannian manifolds. J. Differ. Geom. 2(4), 421–446 (1968)
Zuev, S.A., Sidorenko, A.F.: Continuous models of percolation theory. Theor. Math. Phys. 62(1), 51–58 (1985)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ierene Giardina.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was partially supported by FAPESP Grant 2017/10555 and the third author was partially supported by FAPESP Grants 09/52379-8 and 2012/24086-9.
Rights and permissions
About this article
Cite this article
Coletti, C.F., Miranda, D. & Grynberg, S.P. Boolean Percolation on Doubling Graphs. J Stat Phys 178, 814–831 (2020). https://doi.org/10.1007/s10955-019-02462-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-019-02462-6