Random covers of 1-complexes and the Euler characteristic
Rafał Komendarczyk, Jeffrey K. Pullen · arXiv (Cornell University) · 2012
This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a domain by a random cover. In particular we obtain a general formula for the chance that a collection of finitely many compact connected random sets placed on a 1-complex $X$ has a union equal to $X$. The result is derived under certain topological assumptions on the shape of the covering sets (the cover ought to be {\em good}), 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. Techniques rely on a formulation of the coverage criteria in terms of the Euler characteristic of the nerve complex associated to the random cover.