This talk introduces face and cycle simplicial percolation as models for continuum percolation based on random simplicial complexes in Euclidean space. Face simplicial percolation is defined through infinite sequences of k-simplices sharing a (k-1)-dimensional face. In contrast, cycle simplicial percolation demands the existence of an infinite k-surface, thereby generalizing the lattice notion of plaquette percolation. We discuss the sharp phase transition for face simplicial percolation and derive several relationships between face simplicial percolation, cycle simplicial percolation, and classical vacant continuum percolation. We will also draw connections to a variety of topological models for percolation that have been proposed recently in the literature.
This talk is based on joint work with Daniel Valesin