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Percolation on Homology Generators in Codimension One

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

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.

Notes

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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Copyright information

© Springer Nature Switzerland AG 2020

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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