Extra heads and invariant allocations

Abstract

Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y of \Pi such that translating \Pi by -Y yields a configuration whose law is that of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of \Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of \Gamma if and only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law of \Gamma is product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].Comment: Published at http://dx.doi.org/10.1214/009117904000000603 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stationary Random Measures on Homogeneous Spaces

“…The result extends Theorem 13 in [3] and Theorem 9.1 in [7] (both dealing with α = 1) from R d to general homogeneous spaces. Theorem 8.9.…”

“…The occurence of the sample intensityη in (8.18) is explained by the spatial ergodic theorem, see Proposition 9.1 in [7]. The paper [3] has also results on discrete groups in case α = 1. In case of a general homogeneous space with a diffuse invariant measure µ it might be conjectured that α-balanced partitions exist for all α ≤ 1.…”

“…This partition is assumed to be consistent with the flow {θ g : g ∈ G}. Motivated by recent work in [3] and [2] on allocations to a simple point process, stationary partitions were introduced and studied (in case S = G = R d ) in [7]. Stationary tessellations of classical stochastic geometry (see e.g.…”

“…The actual construction of α-balanced partitions is an interesting topic in its own right. Triggered by [10], the case S = G = R d was discussed in [3,2]. Among many other things it was shown there that α-balanced partitions do actually exist for any α ≤ 1.…”

An easier extra head scheme for the Poisson process on 𝐑ⁿ

“…The question for R n was settled by Holroyd and Peres in [4], where the authors establish an equivalence between extra head schemes and balanced allocation rules, i.e. translation-invariant rules for allocating equal-volume areas of space to the points of a realization.…”