Finite random coverings of one-complexes and the Euler characteristic

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.