Extra heads and invariant allocations

Abstract

Let Πbe an ergodic simple point process on R^d and let Π^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of Πand Π^*; that is, one can select a (random) point Y of Πsuch that translating Πby -Y yields a configuration whose law is that of Π^*. We construct shift couplings in which Y and Π^* are functions of Π, and prove that there is no shift coupling in which Πis a function of Π^*. 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 Π. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let Γbe an ergodic random element of {0,1}^{Z^d} and let Γ^* be Γconditioned on Γ(0)=1. A shift coupling X of Γand Γ^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of Γif and only if the marginal E[Γ(0)] is the reciprocal of an integer. When the law of Γis 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].