3,683 research outputs found
Exponents and bounds for uniform spanning trees in d dimensions
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.Comment: 5 pages. v2: references added. v3: closed form results can be
extended slightly (thanks to C. Tanguy). v4: revisions, and a figure adde
Non-embeddable quasi-residual designs
AbstractWe present a new non-embeddable quasi-residual design which has the same parameters as Bhattacharya's design but which is much easier to describe. Furthermore we give the first example of a non-trivial non-embeddable design on less than 16 points
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