Percolation on Homology Generators in Codimension One

TL;DR

This paper develops a new percolation model based on homology generators in high-dimensional spaces, establishing phase transition thresholds and the uniqueness of infinite hole clusters in a topological setting.

Contribution

It introduces a novel percolation model on homology generators in cubical complexes and provides bounds for critical probabilities and cluster uniqueness.

Findings

Established bounds for the critical probability p_c^{hole}

Proved the existence of a phase transition in the model

Demonstrated the uniqueness of the infinite hole cluster

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 $p_c^{\rm hole}$ 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 $p > p_c^{\rm hole}$, the probability $P_p(x^*\overset{\rm hole}{\longleftrightarrow} y^*)$ that two points in the dual lattice $(\mathbb{Z}^d)^*$ belong to the same hole cluster is uniformly greater than 0.