A poisson allocation of optimal tail

The allocation problem for a d-dimensional Poisson point process is to find a way to partition the space to parts of equal size, and to assign the parts to the configuration points in a measurable, "deterministic" (equivariant) way. The goal is to make the diameter R of the part assigned to a configuration point have fast decay.We present an algorithm for d ≥ 3 that achieves an O(exp(-cR^d)) tail, which is optimal up to c. This improves the best previously known allocation rule, the gravitational allocation, which has an exp(-R^1+o(1)) tail. The construction is based on the Ajtai-Komlós-Tusnády algorithm and uses the Gale-Shapley-Hoffman-Holroyd-Peres stable marriage scheme (as applied to allocation problems).