Dataset for "Covering one point process with another"

Fix a set A as a subset of Euclidean space (for example, a disc or a polygon), and a subset B contained in A (we often consider the case B=A).
Place n points X_1, ..., X_n in A, with the locations chosen independently and uniformly at random. We think of these as "transmitters".
Place another m points Y_1, ..., Y_m in B contained in A, which we think of as "receivers".
At each X_i, place a Euclidean ball of radius r.
We define R_{n,m,k} to be the smallest r such that every receiver Y_j has at least k transmitters within distance r.

In the paper "Covering one point process with another" we proved that if m/n tends to tau as n tends to infinity, then the quantity n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n) (for constants c_1,c_2 which we give in the paper) converges to a random variable (whose distribution we also give). These datasets include large numbers of independent samples of n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n).

The dataset is separated into files, and each file into rows. All the data in a given file are generated using fixed sets A and B, and parameters n, m, d, k. Each row in this given file is a single number: the outcome of an experiment, conducted independently of the other rows. In each experiment we place n points at random locations in A, place m points at random locations in B, calculate R_{n,m,k} as described above (and as detailed formally in the paper) and record the value of n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n) on a row. For the next row, we remove the existing points, and place n points in A, m points in B, etc. for the same n,m, A, B, but with the random points chosen independently of previous experiments.

In probabilisitic terms, the rows of a given file are independent and identically distributed random variables with a common distribution, which is the distribution of n R_{n,m,k}^d - c_1 log(n) - c_2 loglog(n).
The distribution depends on A, B, n, m, d and k. Different files were generated using different choices of A, B, n, m, d and k.

The paper was written by Frankie Higgs, Mathew D. Penrose and Xiaochuan Yang.
We thank Keith Briggs for suggesting the problem and advice on the simulations.