Random \v{C}ech Complexes on Manifolds with Boundary

Abstract

Let $M$ be a compact, unit volume, Riemannian manifold with boundary. In this paper we study the homology of a random \v{C}ech-complex generated by a homogeneous Poisson process in $M$. Our main results are two asymptotic threshold formulas, an upper threshold above which the \v{C}ech complex recovers the $k$-th homology of $M$ with high probability, and a lower threshold below which it almost certainly does not. These thresholds are close together in the sense that they have the same leading term. Here $k$ is positive and strictly less than the dimension $d$ of the manifold. This extends work of Bobrowski and Weinberger in [BW17] and Bobrowski and Oliveira [BO19] who establish similar formulas when $M$ is a torus and, more generally, is closed and has no boundary. We note that the cases with and without boundary lead to different answers: The corresponding common leading terms for the upper and lower thresholds differ being $\log (n) $ when $M$ is closed and $(2-2/d)\log (n)$ when $M$ has boundary; here $n$ is the expected number of sample points. Our analysis identifies a special type of homological cycle, which we call a $\Theta$-like-cycle, which occur close to the boundary and establish that the first order term of the lower threshold is $(2-2/d)\log (n)$.