Sharp Phase Transitions for k-Fold Coverage Using Morse Theory
Probability
2025-10-24 v1 Algebraic Topology
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.
Keywords
Cite
@article{arxiv.2510.20329,
title = {Sharp Phase Transitions for k-Fold Coverage Using Morse Theory},
author = {Yohai Reani and Omer Bobrowski},
journal= {arXiv preprint arXiv:2510.20329},
year = {2025}
}