论文标题
在任意维度中缺乏向后的无限途径以进行第一音阶渗透
Absence of backward infinite paths for first-passage percolation in arbitrary dimension
论文作者
论文摘要
在第一阶段的渗透（FPP）中，人们将非负随机变量（权重）$（t_e）$放在图形的边缘，并研究诱导的加权图度量。我们考虑在$ \ mathbb {z}^d $ for $ d \ geq 2 $上进行fpp，并分析地球学的几何属性，该属性正在优化公制的路径。具体而言，我们解决了大型学术存在的问题，这些问题是无限的路径，其子路是大地测量学的。这是一个著名的猜想，源自弗斯滕伯格的问题，最有力地支持了$ d = 2 $，该问题是连续分发的。举重，A.S。不是大型的。在没有未经证实的假设下，我们在这个问题中提供了这个问题的第一个进展。我们的主要结果是，在两位作者的先前论文中介绍的地理图，均以任何确定性方向A.S.不包含双线路径。结果，可以构建不包含大型虫学的点对上平面大地测量学的随机图。这提供了证据表明，如果存在的话，如果它们存在，则不能以翻译不变的方式作为点对上的大地飞机的限制。
In first-passage percolation (FPP), one places nonnegative random variables (weights) $(t_e)$ on the edges of a graph and studies the induced weighted graph metric. We consider FPP on $\mathbb{Z}^d$ for $d \geq 2$ and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating from a question of Furstenberg and most strongly supported for $d=2$ that for continuously distributed i.i.d. weights, there a.s. are no bigeodesics. We provide the first progress on this question in general dimensions under no unproven assumptions. Our main result is that geodesic graphs, introduced in a previous paper of two of the authors, constructed in any deterministic direction a.s. do not contain doubly-infinite paths. As a consequence, one can construct random graphs of subsequential limits of point-to-hyperplane geodesics which contain no bigeodesics. This gives evidence that bigeodesics, if they exist, cannot be constructed in a translation-invariant manner as limits of point-to-hyperplane geodesics.
