Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12

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dc.creatorBenjamini, Itai
dc.creatorKesten, Harry
dc.creatorPeres, Yuval
dc.creatorSchramm, Oded
dc.date2001-07-19
dc.date2003-02-13
dc.date.accessioned2026-07-07T13:09:07Z
dc.date.available2026-07-07T13:09:07Z
dc.descriptionThe uniform spanning forest (USF) in Z^d is the weak limit of random, uniformly chosen, spanning trees in [-n,n]^d. Pemantle proved that the USF consists a.s. of a single tree if and only if d <= 4. We prove that any two components of the USF in Z^d are adjacent a.s. if 5 <= d <= 8, but not if d >= 9. More generally, let N(x,y) be the minimum number of edges outside the USF in a path joining x and y in Z^d. Then a.s. max{N(x,y) : x,y in Z^d} is the integer part of (d-1)/4. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.
dc.descriptionCurrent version: added some comments regarding related problems and implications, and made some corrections
dc.identifierhttps://arxiv.org/abs/math/0107140
dc.identifierhttp://arxiv.org/abs/math/0107140
dc.identifierAnnals Math.160:465-491,2004
dc.identifierdoi:10.4007/=
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/228631
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject60K35; 60J15
dc.titleGeometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12
dc.typetext

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