Limit theorems for critical faces above the vanishing threshold

TL;DR

This paper studies the limiting behavior of critical faces in ech filtrations over a Poisson point process on a torus, revealing convergence properties as the connection radius shrinks.

Contribution

It establishes the convergence of point processes of critical faces in ech complexes under slow decay of the connection radius, extending understanding of topological phase transitions.

Findings

Convergence of critical face point processes in ech filtrations.

Limit theorems for positive and negative critical faces.

Analysis in the regime where critical faces are rare.

Abstract

We investigate convergence of point processes associated with critical faces for a \v{C}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 $\mathcal M_0$-topology, when the connecting radius of a \v{C}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.