Uniform spanning forest on the integer lattice with drift in one coordinate

TL;DR

This paper studies the structure of the Uniform Spanning Forest on a drifted integer lattice, revealing dimension-dependent properties and establishing bounds and connectivity probabilities using novel techniques.

Contribution

It introduces new methods to analyze USF on drifted lattices, showing dimension-dependent phase transitions and connectivity properties.

Findings

In dimensions 1 and 2, USF consists of a single tree.

In dimensions 3 and higher, USF contains infinitely many trees.

All trees are shown to be one-ended across all dimensions.

Abstract

In this article we investigate the Uniform Spanning Forest ($\mathsf{USF}$) in the nearest-neighbour integer lattice $\mathbf{Z}^{d+1} = \mathbf{Z}\times \mathbf{Z}^d$ with an assignment of conductances that makes the underlying (Network) Random Walk ($\mathsf{NRW}$) drifted towards the right of the first coordinate. This assignment of conductances has exponential growth and decay; in particular, the measure of balls can be made arbitrarily close to zero or arbitrarily large. We establish upper and lower bounds for its Green's function. We show that in dimension $d = 1, 2$ the $\mathsf{USF}$ consists of a single tree while in $d \geq 3,$ there are infinitely many trees. We then show, by an intricate study of multiple $\mathsf{NRW}$s, that in every dimension the trees are one-ended; the technique for $d = 2$ is completely new, while the technique for $d \geq 3$ is a major makeover of the technique for the proof of the same result for the graph $\mathbf{Z}^d.$ We finally establish the probability that two or more vertices are $\mathsf{USF}$-connected and study the distance between different trees.