Extra heads and invariant allocations

Let Π be an ergodic simple point process on Rd, and let Π be its Palm version. Thorisson [13] proved that there exists a shift coupling of Π and Π; that is, one can select a (random) point Y of Π such that translating Π by −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 Π is a function of Π. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of Rd) to the points of Π. The construction is based on the Gale-Shapley stable marriage algorithm [2]. Next, let Γ be an ergodic random element of {0, 1} d , and let Γ be Γ conditioned on Γ(0) = 1. A shift coupling X of Γ and Γ is called an extra head scheme. We show that there exists an extra head scheme which is a function of Γ if and only if the marginal E[Γ0] is the reciprocal of an integer. When the law of Γ is product Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada. [email protected]. http://www.math.ubc.ca/̃ holroyd/. Research funded in part by NSF (USA) Grants DMS–0072398 and CCR–0121555, an NSERC (Canada) Discovery Grant, and by the Center for Pure and Applied Mathematics at UC Berkeley. Departments of Statistics and Mathematics, UC Berkeley, CA 94720, USA. [email protected]. http://stat-www.berkeley.edu/̃ peres. Research funded in part by NSF Grants DMS–0104073, DMS–0244479 and CCR–0121555, and a Miller Professorship at UC Berkeley.