Absence of backward infinite paths for first-passage percolation in arbitrary dimension
Absence of backward infinite paths for first-passage percolation in arbitrary dimension
复制标题
任意维度的第一通道渗透不存在向后无限路径
作者：
Gerandy Brito;M. Damron;Jack Hanson
在首次通过渗流（FPP）中，将非负随机变量（权）$（t_e）$放置在图的边上，研究了导出的权图度量。我们考虑了$mathbb{Z}^d$上的FPP，并分析了测地线的几何性质，测地线是度量的优化路径.具体来说，我们解决的问题存在的bigodeesics，这是双无限路径的子路径是测地线。这是一个著名的猜想起源于一个问题的Furstenberg和最有力的支持$d=2$，连续分布的i.i.d.重量，有A.S.不是双测线。我们在没有未经证实的假设的情况下，提供了关于这个问题的一般维度的第一个进展。我们的主要结果是，测地线图，介绍了在以前的文件中的两个作者，构造在任何确定性的方向。不包含双无限路径。因此，我们可以构造点到超平面测地线的连续极限的随机图，而这些测地线不包含双测地线。这证明了双测地线，如果它们存在，不能以一种不变性的方式构造为点到超平面测地线的极限。
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.