Work

Face and cycle percolation

Year: 2025

Type: article

Abstract: We consider face and cycle percolation as models for continuum percolation based on random simplicial complexes in Euclidean space. More precisely, we consider the Vietoris–Rips with connection radius 1, with respect to Euclidean norm, on a stationary Poisson process in $${\mathbb {R}}^D$$ R D of intensity $$\lambda $$ λ . Face percolation is defined through infinite sequences of d-simplices sharing a $$(d-1)$$ ( 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.

Source: Journal of Applied and Computational Topology

DOI: 10.1007/s41468-025-00202-2

Authors: Christian Hirsch, Daniel Valesin

Institutions: Aarhus University, University of Warwick

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,390

Funders: Aarhus Universitet, Aarhus Universitets Forskningsfond