2018/09/20 by Yasuaki Hiraoka, Tatsuya Mikami, Hiraoka, Yasuaki +1
Computer Science · Mathematics · Physics and Astronomy · #52C99 #60D05 #82B43 #Algebraic Topology (math.AT) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1809.07490
openalex publication_date 2018/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝd 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 pc\rm 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, when p > pc\rm hole, the probability Pp(x^*\overset\rm hole\longleftrightarrow y^*) that two points in the dual lattice (ℤd)^* belong to the same hole cluster is uniformly greater than 0.