Sharp Phase Transitions for k-Fold Coverage Using Morse Theory

Yohai Reani; Omer Bobrowski

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arXiv · 2510.20329

Sharp Phase Transitions for k-Fold Coverage Using Morse Theory

Yohai Reani·Omer Bobrowski

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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.

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Yohai Reani, Omer Bobrowski. 2025-10-23. Sharp Phase Transitions for k-Fold Coverage Using Morse Theory. https://arxiv.org/abs/2510.20329

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