Abstract
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.
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Notes
- 1.
A finite-dimensional cylinder set is a set \(\{\omega \in \Omega : \omega _{e_i} = \epsilon _i, i = 1,2, \ldots ,n\}\) for some \(n \in \mathbb {N}\), \(e_1, \ldots , e_n \in {\mathbb {E}^d}\) and 𝜖 i ∈{0, 1}.
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Acknowledgements
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.
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Hiraoka, Y., Mikami, T. (2020). Percolation on Homology Generators in Codimension One. In: Baas, N., Carlsson, G., Quick, G., Szymik, M., Thaule, M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-43408-3_12
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