Document Type

Article

Publication Date

2010

Abstract

Let P be a Poisson process of intensity one in the infinite plane R2. We surround each point x of P by the open disc of radius r centred at x. Now let Sn be a fixed disc of area n, and let Cr(Sn) be the set of discs which intersect Sn. Write Erk for the event that Cr(Sn) is a k-cover of Sn, and Frk for the event that Cr(Sn) may be partitioned into k disjoint single covers of Sn. We prove that P(ErkFrk) ≤ ck / logn, and that this result is best possible. We also give improved estimates for P(Erk). Finally, we study the obstructions to k-partitionability in more detail. As part of this study, we prove a classification theorem for (deterministic) covers of R2 with half-planes that cannot be partitioned into two single covers.

Publication Title

Advances in Applied Probability

Volume

42

Issue

1

First Page

1

Last Page

25

Required Publisher's Statement

Published by Project Euclid

DOI: 10.1239/aap/1269611141

Comments

This is the authors' version of the paper. The publisher's version is here: http://projecteuclid.org/euclid.aap/1269611141

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Mathematics Commons

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