Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape
Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape
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DOI：
10.1214/21-ejp706
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发表时间：
2019-12
影响因子：
1.4
通讯作者：
E. Bates;S. Ganguly;A. Hammond
中科院分区：
文献类型：
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作者：
E. Bates;S. Ganguly;A. Hammond
在kardar - paris - zhang普适类中，时空Airy表被推测为最后通道渗流模型的标准尺度极限。在Dauvergne， Ortmann和Vir\'ag最近的工作arXiv:1812.00309中，这个天体被构造出来，并被证明是一个这样的模型：布朗最后通道渗透的抛物线修正后的极限。这个极限对象，称为有向景观，允许任意两个时空点$(x,s)$和$(y,t)$之间的测地线路径，且$s<t$。在这篇文章中，我们研究了这些路径集合的分形性质。我们的主要结果涉及允许不相交测地线的特殊端点。首先，我们确定两个不同的起始位置$x_1$和$x_2$，并考虑测地线从$(x_1,0)\到（y,1）$和$(x_2,0)\到（y,1）$。我们证明了只有在时间$1$时测地线才会合的集合$y\in\mathbb{R}$具有1 / 2的豪斯多夫维数。其次，我们考虑端点$(x,0)$和$(y,1)$之间存在两个测地线，它们只在$0$和$1$相交。我们证明了这样的$(x,y)\in\mathbb{R}^2$的集合也具有1 / 2的豪斯多夫维数。证明需要几个独立的输入，包括(i)与arXiv:1904.01717中研究的所谓的差权剖面的联系；（ii）在小间隔内开始和结束的不相交测地线数目的尾部估计。后一个结果推广了arXiv:1709.04110对预极限模型所证明的类似估计。
Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work arXiv:1812.00309 of Dauvergne, Ortmann, and Vir\'ag, this object was constructed and shown to be the limit after parabolic correction of one such model: Brownian last passage percolation. This limit object, called the directed landscape, admits geodesic paths between any two space-time points $(x,s)$ and $(y,t)$ with $s<t$. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations $x_1$ and $x_2$, and consider geodesics traveling $(x_1,0)\to (y,1)$ and $(x_2,0)\to (y,1)$. We prove that the set of $y\in\mathbb{R}$ for which these geodesics coalesce only at time $1$ has Hausdorff dimension one-half. Second, we consider endpoints $(x,0)$ and $(y,1)$ between which there exist two geodesics intersecting only at times $0$ and $1$. We prove that the set of such $(x,y)\in\mathbb{R}^2$ also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in arXiv:1904.01717; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in arXiv:1709.04110.