The Green's function on the double cover of the grid and application to the uniform spanning tree trunk

TL;DR

This paper calculates the Green's function on a double cover of the square lattice to analyze local statistics of the uniform spanning tree trunk and triple points, revealing algebraic structures of the probabilities involved.

Contribution

It introduces a method to compute the Green's function on a double cover of the lattice and applies it to determine local statistics of the uniform spanning tree trunk and triple points.

Findings

Probabilities of cylinder events are in {}Q[2} for the trunk.

Probabilities of triple points are in {}Q[1/] for large-scale branching.

Method reduces the problem to a dimer system with isolated monomers.

Abstract

We compute the Green's function on the double cover of ${\mathbb Z}^2$, branched over a vertex or a face. We use this result to compute the local statistics of the "trunk" of the uniform spanning tree on the square lattice, i.e., the limiting probabilities of cylinder events conditional on the path connecting far away points passing through a specified edge. We also show how to compute the local statistics of large-scale triple points of the uniform spanning tree, where the trunk branches. The method reduces the problem to a dimer system with isolated monomers, and we compute the inverse Kasteleyn matrix using the Green's function on the double cover of the square lattice. For the trunk, the probabilities of cylinder events are in ${\mathbb Q}[\sqrt{2}]$, while for the triple points the probabilities are in ${\mathbb Q}[1/\pi]$.