Covering a compact space by fixed-radius or growing random balls

David J. Aldous

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arXiv · 2101.12671

Covering a compact space by fixed-radius or growing random balls

David J. Aldous

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Abstract

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a "weak concentration" bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the former concerning coverage by i.i.d. random sets with arbitrary distribution, and the latter concerning hitting times for Markov chains with a strong monotonicity property. The growth model seems generally more tractable, and we record some basic results and open problems for that model.

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David J. Aldous. 2021-01-29. Covering a compact space by fixed-radius or growing random balls. https://arxiv.org/abs/2101.12671

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