Homological percolation and the Euler characteristic.

Abstract

In this paper we study the connection between the zeros of the expected Euler characteristic curve and the phenomenon which we refer to as homological percolation-the formation of "giant" cycles in persistent homology, which is intimately related to classical notions of percolation. We perform an experimental study that covers four different models: site percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields. All the models are generated on the flat torus T^{d} for d=2,3,4. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homological percolation. Our results also provide some insight about the approximation error. Further study of this connection could have powerful implications both in the study of percolation theory and in the field of topological data analysis.