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)].


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