Random Čech complexes on manifolds with boundary. Issue 2 (20th January 2022) - de Kergorlay, Henry‐Louis; Tillmann, Ulrike; Vipond, Oliver
Abstract: 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 k th 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.