The Annals of Probability

Extra heads and invariant allocations

Alexander E. Holroyd and Yuval Peres

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Abstract

Let Π be an ergodic simple point process on ℝd and let Π* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057–2064] 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 [Amer. Math. Monthly 69 (1962) 9–15]. 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[Γ(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 EexpcXd<∞; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405–1425].

Article information

Source
Ann. Probab., Volume 33, Number 1 (2005), 31-52.

Dates
First available in Project Euclid: 11 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1108141719

Digital Object Identifier
doi:10.1214/009117904000000603

Mathematical Reviews number (MathSciNet)
MR2118858

Zentralblatt MATH identifier
1097.60032

Subjects
Primary: 60G55: Point processes 60K60

Keywords
Shift coupling point process Palm process invariant transport invariant allocation

Citation

Holroyd, Alexander E.; Peres, Yuval. Extra heads and invariant allocations. Ann. Probab. 33 (2005), no. 1, 31--52. doi:10.1214/009117904000000603. https://projecteuclid.org/euclid.aop/1108141719


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References

  • Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29--66.
  • Gale, D. and Shapley, L. (1962). College admissions and stability of marriage. Amer. Math. Monthly 69 9--15.
  • Häggström, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423--1436.
  • Hoffman, C., Holroyd, A. E. and Peres, Y. (2004). A stable marriage of Poisson and Lebesgue. Unpublished manuscript.
  • 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.
  • Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Comm. Probab. 8 17--27.
  • Kallenberg, O. (2002). Foundations of Modern Probability. Probability and Its Applications, 2nd ed. Springer, New York.
  • 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.
  • Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press.
  • Meshalkin, L. D. (1959). A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk. SSSR 128 41--44.
  • Talagrand, M. (1994). The transportation cost from the uniform measure to the empirical measure in dimension $\ge3$. Ann. Probab. 22 919--959.
  • Thorisson, H. (1995). On time- and cycle-stationarity. Stochastic Process. Appl. 55 183--209.
  • Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Probab. 24 2057--2064.