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