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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)
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[1] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29–66. · Zbl 0924.43002
[2] Gale, D. and Shapley, L. (1962). College admissions and stability of marriage. Amer. Math. Monthly 69 9–15. · Zbl 0109.24403
[3] Häggström, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423–1436. · Zbl 0895.60098
[4] Hoffman, C., Holroyd, A. E. and Peres, Y. (2004). A stable marriage of Poisson and Lebesgue. Unpublished manuscript. · Zbl 1111.60008
[5] Holroyd, A. E. and Liggett, T. M. (2001). How to find an extra head: Optimal random shifts of Bernoulli and Poisson random fields. Ann. Probab. 29 1405–1425. · Zbl 1019.60048
[6] Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Comm. Probab. 8 17–27. · Zbl 1060.60048
[7] Kallenberg, O. (2002). Foundations of Modern Probability . Probability and Its Applications , 2nd ed. Springer, New York. · Zbl 0996.60001
[8] Liggett, T. M. (2002). Tagged particle distributions or how to choose a head at random. In In and Out of Equilibrium (V. Sidoravicious, ed.) 133–162. Birkhäuser, Boston. · Zbl 1108.60319
[9] Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces . Cambridge Univ. Press. · Zbl 0819.28004
[10] Meshalkin, L. D. (1959). A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk. SSSR 128 41–44. · Zbl 0099.12301
[11] Talagrand, M. (1994). The transportation cost from the uniform measure to the empirical measure in dimension \(\geq3\). Ann. Probab. 22 919–959. JSTOR: · Zbl 0809.60015
[12] Thorisson, H. (1995). On time- and cycle-stationarity. Stochastic Process. Appl. 55 183–209. · Zbl 0817.60050
[13] Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Probab. 24 2057–2064. · Zbl 0879.60051
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