Extra heads and invariant allocations

By Alexander E. Holroyd, Yuval Peres. 2003

83 citations recorded by the source corpus.

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].

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6b6338f8e3fcaa0b7efe3f334c4b490dd2376788

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Citations in the corpus

Invariant transports of stationary random measures and mass-stationarity (2009, methodology): “Another special case of ψ are the transport rules of [11] arising in case ξ is the Lebesgue measure on G = R .”
A Stable Marriage of Poisson and Lebesgue (2005, background): “By Christopher Hoffman,1 Alexander E. Holroyd2 and Yuval Peres3”
Poisson Matching (2007, methodology): “Now let F̃n be Fn conditioned on the event Gn (this corrects a minor error in [14]).”
Lectures on the Poisson Process (2017, methodology): “Stable allocations balancing Lebesgue measure and the stationary Poisson process were introduced and studied by Holroyd and Peres [49].”
Gravitational allocation to Poisson points (2006, background): “In [8] it was shown that if ψZ is an allocation rule then T = TZ = ψZ(0) is a non-randomized extra head rule, and conversely, given a non-randomized extra head rule TZ , the mapping ψZ(x) = TZ−x is an allocation rule.”
Optimal transport from Lebesgue to Poisson (2010, methodology): “[HP05] A. E. Holroyd and Y. Peres.”