The nerd snipers problem

Abstract What fraction of points in a planar Poisson process are not the nearest neighbor of any other point? That is, what is the probability that a given particle is not the nearest neighbor of any other particle in a classical two-dimensional ideal gas model? In 1987, Tao and Wu presented an award-winning tour de force of integration to solve this problem, but in the time since, errors were discovered in their solution, and the technical difficulty precluded researchers from correcting them. In this paper, we rectify this situation by providing a complete solution to the original problem. First, we prove a correct version of Tao and Wu’s intended change of variables, and in doing so, we identify an important constraint that was missing in one of the integration variables. Next, we use ideas from Fourier–Motzkin elimination to prove bounds of integration that correctly incorporate this missing constraint. The result is a weighted sum of two double integrals, four quadruple integrals, four sextuple integrals, and two octuple integrals, and we apply a variety of techniques to numerically compute each of these integrals. Considering the error-riddled history of this problem, we conclude by discussing the various ways we verified different portions of our solution.