Connected Allocation to Poisson Points in R

This note answers one question in [1], concerning the connected allocation for the Poisson process in R. The proposed solution makes use of the Riemann map from the plane minus the minimal spanning forest of the Poisson point process to the halfplane. A picture of a numerically simulated example is included. 1 The problem Let X ⊂ R be a discrete set. We call the elements of X centers, the elements of R – sites, and we write L for the Lebesgue measure in R. An allocation of R to X with appetite α ∈ [0,∞] is a measurable function ψ : R → X ∪ {∞,∆} such that Lψ(ξ) ≤ α for all ξ ∈ X and Lψ(∆) = 0. We call ψ(ξ) the territory of the center ξ. A center ξ is sated if Lψ(ξ) = α and unsated otherwise. A site x is claimed if ψ(x) ∈ X and unclaimed if ψ(x) = ∞. The allocation is undefined at x if ψ(x) = ∆. One particular question we’re interested in is the following [1]: Is there a translation-equivariant allocation for the Poisson process of unit intensity in the critical two-dimensional case (d = 2 and α = 1), in which every territory is connected? The goal of the present note is to describe one such allocation. Note: for a far more general approach to this question in d ≥ 3 see [3]. 2 Construction of a connected allocation Let X be a realization of Poisson process (of intensity 1) in R. Let the minimal spanning forest T be the union of edges e = (x, y), x, y ∈ X , such that there is no path from x to y in the complete graph on X with all edges strictly shorter than e (see [4] for other definitions and references on the subject). For a 2-dimensional Poisson process Alexander [2] proved that T is a.s. a tree with one topological end (a topological end in a graph is a class of equivalence of semi-infinite paths modulo finite