# TAILIEUCHUNG - Đề tài " Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12, . . . "

## The uniform spanning forest (USF) in Zd is the weak limit of random, uniformly chosen, spanning trees in [−n, n]d . Pemantle [11] proved that the USF consists . of a single tree if and only if d ≤ 4. We prove that any two components of the USF in Zd are adjacent . 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 Zd . Then max N (x, y) : x, y ∈ Zd = (d − 1)/4 . | Annals of Mathematics Geometry of the uniform spanning forest Transitions in dimensions 4 8 12 . By Itai Benjamini Harry Kesten Yuval Peres and Oded Schramm Annals of Mathematics 160 2004 465 491 Geometry of the uniform spanning forest Transitions in dimensions 4 8 12 . By Itai Benjamini HARRy Kesten Yuval Peres and Oded Schramm Abstract The uniform spanning forest USF in Zd is the weak limit of random uniformly chosen spanning trees in n n d. Pemantle 11 proved that the USF consists . of a single tree if and only if d 4. We prove that any two components of the USF in Zd are adjacent . 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 Zd. Then max N x y x y e Zd _ d 1 4_ . The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof. 1. Introduction A uniform spanning tree UST in a finite graph is a subgraph chosen uniformly at random among all spanning trees. A spanning tree is a subgraph such that every pair of vertices in the original graph are joined by a unique simple path in the subgraph. The uniform spanning forest USF in Zd is a random subgraph of Zd that was defined by Pemantle 11 following a suggestion of R. Lyons as follows The USF is the weak limit of uniform spanning trees in larger and larger finite boxes. Pemantle showed that the limit exists that it does not depend on the sequence of boxes and that every connected component of the USF is an infinite tree. See Benjamini Lyons Peres and Schramm 2 denoted BLPS below for a thorough study of the construction and properties of the USF as well as references to other works on the subject. Let T x denote the tree in the USF which contains the vertex x. Research partially supported by NSF grants DMS-9625458 Kesten and DMS-9803597 Peres and by a Schonbrunn Visiting Professorship Kesten . Key words and phrases. Stochastic dimension Uniform spanning forest. 466 ITAI BENJAMINI HARRY .

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