Homological Percolation and the Euler Characteristic

TL;DR

This paper investigates the relationship between homological percolation and the zeros of the expected Euler characteristic curve across various models on flat tori, revealing that these zeros approximate critical percolation thresholds.

Contribution

It establishes an experimental link between the zeros of the Euler characteristic curve and homological percolation thresholds across multiple models and dimensions.

Findings

Zeros of the Euler characteristic curve approximate critical percolation values

Simulation results support the connection across different models and dimensions

Provides insights into the approximation error of the method

Abstract

In this paper we study the connection between the phenomenon of homological percolation (the formation of "giant" cycles in persistent homology), and the zeros of the expected Euler characteristic curve. 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.