Random Čech complexes on manifolds with boundary

Let M be a compact, unit volume, Riemannian manifold with boundary. We study the homology of a random Čech‐complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the kth homology of M with high probability, and a lower threshold below which it almost certainly does not. These thresholds share the same leading term. This extends work of Bobrowski–Weinberger and Bobrowski–Oliveira who establish similar formulas when M has no boundary. The cases with and without boundary differ: 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 occurring close to the boundary.