Extra heads and invariant allocations

• 成果类型: Article
• 署名作者: Holroyd, AE; Peres, Y
• 署名单位: University of British Columbia; University of California System; University of California Berkeley; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000603
• 发表日期: 2005
• 页码: 31-52
• 关键词: percolation; bernoulli

摘要

Let Pi be an ergodic simple point process on R-d and let Pi* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] 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 Rd) to the points of Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer Math. Monthly 69 (1962) 9-15]. Next, let Gamma be an ergodic random element of {0, 1}(Zd) 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 E[Gamma(0)] is the reciprocal of an integer. When the law of Gamma is product measure and d greater than or equal to 3, we prove that there exists an extra head scheme X satisfying E exp c cparallel toXparallel to(d) < infinity; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].