  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.
