论文标题
在整数晶格上跨越森林，在一个坐标中漂移
Uniform spanning forest on the integer lattice with drift in one coordinate
论文作者
论文摘要
在本文中，我们调查了最近的跨森林（$ \ mathsf {usf} $），在最近的neighbour整数晶格中（$ \ Mathsf {nrw} $）向第一个坐标的右侧漂移。这种电导的分配具有指数的增长和衰减。特别是，可以任意将球的度量衡量接近零或任意大。我们为其绿色功能建立上限和下限。我们表明，在尺寸中$ d = 1，2 $ $ \ mathsf {usf} $由一棵树组成，而在$ d \ geq 3中，$有很多树。然后，我们通过对多个$ \ mathsf {nrw} $ s的复杂研究表明，在每个维度上，树是一个末端； $ d = 2 $的技术是全新的，而$ d \ geq 3 $的技术是该技术的重大改造，以证明图形$ \ mathbf {z}^d。$，我们最终确定了两个或多个顶点是$ \ mathsf {usf {usf} $ - 连接和研究不同树之间的两个或更多顶点。
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.
