## Extra heads and invariant allocations.(English)Zbl 1097.60032

Summary: Let $$\Pi$$ be an ergodic simple point process on $$\mathbb R^d$$ and let $$\Pi^*$$ be its Palm version. H. Thorisson [Ann. Probab. 24, No. 4, 2057–2064 (1996; Zbl 0879.60051)] 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 $$\mathbb R^d$$) to the points of $$\Pi$$. The construction is based on the Gale-Shapley stable marriage algorithm [see D. Gale and L. S. Shapley, Am. Math. Mon. 69, 9–15 (1962; Zbl 0109.24403)]. Next, let $$\Gamma$$ be an ergodic random element of $$\{0,1\}^{\mathbb Z^d}$$ 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 $${\mathbf E}[\Gamma(0)]$$ 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 $${\mathbf E}\exp c\|X\|^d<\infty$$; this answers a question of A. E. Holroyd and T. M. Liggett [Ann. Probab. 29, No. 4, 1405–1425 (2001; Zbl 1019.60048)].

### MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

### Keywords:

shift coupling; point process; Palm process; invariant transport

### Citations:

Zbl 0879.60051; Zbl 0109.24403; Zbl 1019.60048
Full Text:

### References:

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