Finite random coverings of one-complexes and the Euler characteristic

DOI10.12775/TMNA.2015.008zbMATH Open1360.55018arXiv1207.1133OpenAlexW3100322773MaRDI QIDQ519442

Jeffrey Pullen, R. Komendarczyk

Publication date: 4 April 2017

Published in: Topological Methods in Nonlinear Analysis (Search for Journal in Brave)

Abstract: This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a one dimensional domain $X$ by a random covering, and develops techniques applicable to the problem beyond the one dimensional case. In particular we obtain a general formula for the chance that a collection of finitely many compact connected random sets placed on $X$ has a union equal to $X$. The result is derived under certain topological assumptions on the shape of the covering sets (the covering ought to be {em good}, which holds if the diameter of the covering elements does not exceed a certain size), but no a priori requirements on their distribution. An upper bound for the coverage probability is also obtained as a consequence of the concentration inequality. The techniques rely on a formulation of the coverage criteria in terms of the Euler characteristic of the nerve complex associated to the random covering.

Full work available at URL: https://arxiv.org/abs/1207.1133

zbMATH Keywords

Vietoris-Rips complexnervesrandom complexescomplete coverage probability

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) General topology of complexes (57Q05) Simplicial sets and complexes in algebraic topology (55U10) Singular homology and cohomology theory (55N10)

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