Extra heads and invariant allocations

Let n be an ergodic simple point process on E d and let n* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of Π and n*; that is, one can select a (random) point Y of n such that translating n 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 n is 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 n. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let Γ be an ergodic random element of {0, 1} Zd and let Γ* be r conditioned on r(0) = 1. A shift coupling X of r and Γ* is called an extra head scheme. We show that there exists an extra head scheme which is a function of r if and only if the marginal E[r(0)] is the reciprocal of an integer. When the law of Γ is product measure and d > 3, we prove that there exists an extra head scheme X satisfying E exp c∥X∥ d < ∞; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].