Sharp Phase Transitions for k-Fold Coverage Using Morse Theory

TL;DR

This paper uses Morse theory to analyze phase transitions in random k-coverage, establishing sharp thresholds for coverage and the distribution of uncovered regions, advancing understanding of geometric coverage problems.

Contribution

It introduces a novel Morse theory-based framework to study phase transitions in k-coverage, providing sharp thresholds and Poisson approximations for uncovered regions.

Findings

Sharp phase transition for the number of critical points of the k-NN distance function.

Phase transition for k-coverage established.

Poisson process approximation for last uncovered regions.

Abstract

We introduce a novel approach for studying random k-coverage, using Morse theory for the k-nearest neighbor (k-NN) distance function. We prove a sharp phase transition for the number of critical points of the k-NN distance function, from which we conclude a phase transition for k-coverage. In addition, in the critical window our new framework enables us to prove a Poisson process approximation (in both location and size) for the last uncovered regions.