Random Čech complexes on manifolds with boundary

Journal article

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.

Authors: de Kergorlay, H-L; Tillmann, U; Vipond, O

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Random Structures and Algorithms More from this journal
• Volume: 61
• Issue: 2
• Pages: 309-352
• Publication date: 2022-01-20
• Acceptance date: 2021-04-13
• DOI: 10.1002/rsa.21062
• EISSN: 1098-2418
• ISSN: 1042-9832
• Language: English
• Keywords: FFR; random geometric complexes; stochastic topology; homology