论文信息 - Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12

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

The 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 = 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.

Harry Kesten | Yuval Peres | Itai Benjamini | Oded Schramm

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