Percolation on Homology Generators in Codimension One

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 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 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, the probability Pp(x∗↔ hole y∗) that two points in the dual lattice belong to the same hole cluster is uniformly greater than 0.