Limit theorems for critical faces above the vanishing threshold

We investigate convergence of point processes associated with critical faces for a ech filtration built over a homogeneous Poisson point process in the d-dimensional flat torus.The convergence of our point process is established in terms of the M 0topology, when the connecting radius of a ech complex decays to 0, so slowly that critical faces are even less likely to occur than those in the regime of threshold for homological connectivity.We also obtain a series of limit theorems for positive and negative critical faces, all of which are considerably analogous to those for critical faces.