Percolation on Homology Generators in Codimension One

  • Yasuaki Hiraoka
  • Tatsuya MikamiEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 15)


This paper introduces a new percolation model motivated from polymer materials. The mathematical model is defined over a random cubical set in the d-dimensional space \(\mathbb {R}^d\) and focuses on generations and percolations of (d − 1)-dimensional holes as higher dimensional topological objects. Here, the random cubical set is constructed by the union of unit faces in dimension d − 1 which appear randomly and independently with probability p, and holes are formulated by the homology generators. Under this model, the upper and lower estimates of the critical probability \(p_c^{\operatorname {hole}}\) of the hole percolation are shown in this paper, implying the existence of the phase transition. The uniqueness of infinite hole cluster is also proven. This result shows that, in the supercritical phase, \(p > p_c^{\operatorname {hole}}\), the probability \(P_p(x^* \overset {\mathrm {hole}}{\longleftrightarrow } y^*)\) that two points in the dual lattice \((\mathbb {Z}^d)^*\) belong to the same hole cluster is uniformly greater than 0.



The authors would like to thank Tomoyuki Shirai, Kenkichi Tsunoda and Masato Takei for their valuable suggestions and useful discussions. This work is partially supported by JST CREST Mathematics 15656429 and JSPS Grant-in-Aid for challenging Exploratory Research 17829801.


  1. 1.
    Aizenman, M., Chayes, J., Chayes, L., Frőhlich, J., Russo, L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Comm. Math. Phys. 92, 19–69 (1983)Google Scholar
  2. 2.
    Aizenman, M., Kesten, H., Newman, C.: Uniqueness of the infinite cluster and continuity of connectivity functions for short- and long-range percolation, Comm. Math. Phys. 92, 505–532 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bobrowski, O., Kahle, M.: Topology of random geometric complexes: a survey. J. Appl. Comput. Topology.
  4. 4.
    Erdős, P., Rényi, A.: On the Evolution of Random Graphs. Publ. Math. Inst. Hungarian Acad. Sci. 5A, 17–61 (1960)Google Scholar
  5. 5.
    Fitzner, R., Hofstad, R.: Mean-field behavior for nearest-neighbor percolation in d > 10, Electron. J. Probab. 22, no. 43, 1–65 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Grimmett, G., Holroyd, A., Kozma, G.: Percolation of finite clusters and infinite surfaces, Math. Proc. Cambridge Philos. Soc. 156, no. 2, 263–279 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grimmett, G., Holroyd, A.: Plaquettes, Spheres, and Entanglement. Electron. J. Probab. 15, 1415–1428 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grimmett, G.: Percolation. Springer-Verlag, Berlin (1999)CrossRefGoogle Scholar
  9. 9.
    Harris, T.: A lower bound for the critical probability in a certain percolation process. Math. Proc. Camb. Philos. Soc. 56, 13–20 (1960)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hiraoka, Y., Tsunoda, K.: Limit theorems for random cubical homology. Dicrete Comput. Geom. 60, 665–687 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ichinomiya, T., Obayashi, I., Hiraoka, Y.: Persistent homology analysis of craze formation. Phys. Rev. E. 95, 012504 (2017)CrossRefGoogle Scholar
  12. 12.
    Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Springer-Verlag, New York (2004)CrossRefGoogle Scholar
  13. 13.
    Kahle, M.: Topology of random simplicial complexes: a survey. In: Algebraic Topology: applications and new directions. Contemp. Math. 620 (Tillmann, U., Galatius, S., Sinha, D. eds.). pp. 201–221. Amer. Math. Soc., Providence (2014)Google Scholar
  14. 14.
    Kesten, H.: The critical probability of bond percolation on the square lattice equals \(\frac {1}{2}\). Comm. Math. Phys. 74, 41–59 (1980)Google Scholar
  15. 15.
    Meester, R., Roy, R.: Continuum Percolation. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  16. 16.
    Menshikov, M.: Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR. 288(6), 1308–1311 (1986)MathSciNetGoogle Scholar
  17. 17.
    Werman, M., Wright, M.L.: Intrinsic volumes of random cubical complexes. Discrete Comput. Geom. 56, 93–113 (2016)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Center for Advanced StudyInstitute for the Advanced Study of Human Biology (WPI-ASHBi), Kyoto University Institute for Advanced Study, Kyoto UniversityKyotoJapan
  2. 2.Center for Advanced Intelligence ProjectRIKENTokyoJapan
  3. 3.Department of MathematicsFaculty of Science, Kyoto UniversityKyotoJapan

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