Extra heads and invariant allocations
| dc.creator | Holroyd, Alexander E. | |
| dc.creator | Peres, Yuval | |
| dc.date | 2003-06-27 | |
| dc.date | 2005-05-20 | |
| dc.date.accessioned | 2026-07-07T04:59:15Z | |
| dc.date.available | 2026-07-07T04:59:15Z | |
| dc.description | Let Πbe an ergodic simple point process on R^d and let Π^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of Πand Π^*; that is, one can select a (random) point Y of Πsuch that translating Πby -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 Πis 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 Π. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let Γbe an ergodic random element of {0,1}^{Z^d} and let Γ^* be Γconditioned on Γ(0)=1. A shift coupling X of Γand Γ^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of Γif and only if the marginal E[Γ(0)] is the reciprocal of an integer. When the law of Γis product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425]. | |
| dc.description | Published at http://dx.doi.org/10.1214/009117904000000603 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0306402 | |
| dc.identifier | http://arxiv.org/abs/math/0306402 | |
| dc.identifier | Annals of Probability 2005, Vol. 33, No. 1, 31-52 | |
| dc.identifier | doi:10.1214/009117904000000603 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67908 | |
| dc.subject | Probability | |
| dc.subject | 60G55, 60K60 (Primary) | |
| dc.title | Extra heads and invariant allocations | |
| dc.type | text |