Exponents and bounds for uniform spanning trees in d dimensions

TL;DR

This paper investigates uniform spanning trees in d-dimensional lattices, deriving exact exponents that bound the decay of probabilities related to the tree structure using Grassmann integrals and Laplacian matrices.

Contribution

It introduces a novel method to compute bounds on decay exponents for spanning trees in high dimensions using Grassmann integrals.

Findings

Derived exact bounds for decay exponents in d-dimensional spanning trees.

Connected Laplacian matrices with probabilistic properties of spanning trees.

Provided analytical tools for studying large lattice limits.

Abstract

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.