Face and cycle percolation

TL;DR

This paper studies continuum percolation models based on random simplicial complexes, analyzing phase transitions and comparing critical intensities for face and cycle percolation in Euclidean space.

Contribution

It introduces and compares face and cycle percolation models, establishing phase transition properties and inequalities between their critical intensities.

Findings

Sharp phase transition for face percolation.

Comparison results between critical intensities.

Inequality involving critical intensities of alternative simplex percolation.

Abstract

We consider face and cycle percolation as models for continuum percolation based on random simplicial complexes in Euclidean space. Face percolation is defined through infinite sequences of $d$-simplices sharing a $(d-1)$-dimensional face. In contrast, cycle percolation demands the existence of infinite $d$-cycles, thereby generalizing the lattice notion of plaquette percolation. We discuss the sharp phase transition for face percolation and derive comparison results between the critical intensities for face and cycle percolation. Finally, we consider an alternate version of simplex percolation, by declaring simplices to be neighbors whenever they are sufficiently close to each other, and prove a strict inequality involving the critical intensity of this alternate version and that of face percolation.