Random Čech complexes on manifolds with boundary. Issue 2 (20th January 2022)

• Record Type: Journal Article
• Authors: de Kergorlay, Henry‐Louis; Tillmann, Ulrike; Vipond, Oliver
• Is Part Of: Random structures & algorithms. Volume 61:Issue 2(2022)
• Page Start: 309
• Page End: 352
• Publication Date: 2022-01-20
• Subjects: homology -- random geometric complexes -- stochastic topology; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.21062
• Languages: English
• ISSNs: 1042-9832

Abstract

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.