Exponents and bounds for uniform spanning trees in d dimensions

Details Exponents and bounds for uniform spanning trees in d dimensions Exponents and bounds for uniform spanning trees in d dimensions

2004-02-20

arxiv.org/abs/cond-mat/0402533v4

Phys. Rev. E 70, 027103 (2004)

Physics

Condensed Matter

Statistical Mechanics

5 pages. v2: references added. v3: closed form results can be extended slightly (thanks to C. Tanguy). v4: revisions, and a fi

Scientific paper

10.1103/PhysRevE.70.027103

Uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph (such as a portion of a cubic lattice in d dimensions), with equal probability for each distinct tree. Some properties of such trees can be obtained in terms of the Laplacian matrix on the graph, by using Grassmann integrals. We use this to obtain exact exponents that bound those for the power-law decay of the probability that k distinct branches of the tree pass close to each of two distinct points, as the size of the lattice tends to infinity.

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