Finite random coverings of one-complexes and the Euler characteristic

TL;DR

This paper develops an algebraic topology approach to compute the probability that a random collection of sets covers a one-dimensional domain, providing a general formula and bounds based on the Euler characteristic.

Contribution

It introduces a novel algebraic topology method to determine coverage probabilities for random coverings of one-complexes, extending to broader cases beyond one dimension.

Findings

Derived a general formula for coverage probability using Euler characteristic.

Established an upper bound for coverage probability via concentration inequalities.

Provided topological conditions under which the formula applies.

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.