Covering one point process with another

Let X1, X2, . . . and Y1, Y2, . . . be i.i.d. random uniform points in a bounded domain A ⊂ R 2 with smooth or polygonal boundary. Given n, m, k ∈ N, define the two-sample k-coverage threshold Rn,m,k to be the smallest r such that each point of {Y1, . . . , Ym} is covered at least k times by the disks of radius r centred on X1, . . . , Xn. We obtain the limiting distribution of Rn,m,k as n → ∞ with m = m(n) ∼ τn for some constant τ > 0, with k fixed. If A has unit area, then nπR2 n,m(n),1 − log n is asymptotically Gumbel distributed with scale parameter 1 and location parameter log τ . For k > 2, we find that nπR2 n,m(n),k − log n − (2k − 3) log log n is asymptotically Gumbel with scale parameter 2 and a more complicated location parameter involving the perimeter of A; boundary effects dominate when k > 2. For k = 2 the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all k.