论文标题
通过构造当地时代
Fractal geometry of the space-time difference profile in the directed landscape via construction of geodesic local times
论文作者
论文摘要
有向景观是飞机上的一个随机指标（分别称为空间和时间坐标），在Dauvergne，Ortmann和Virág的突破性工作中构建，此后已被证明是percolation contrcolation central-a kardar-Paris-par-s-Pare-s-s-s-s-a ar-par-par-s-s-kardar-Parii-s-a ar-parii-z ar-s-s-a ar-parii-z ar-s-s centencyii-a kardar-parii-z ar-s-z ar-s-z ar-z ar-s-z的范围的缩放限制。它表现出几种规模的不变特性，使其成为丰富的分形行为的自然来源。这项研究是在Basu-Ganguly-Hammond启动的，其中考虑了两个固定点（例如$（\ pm 1,0）$）的差异分布的差异。由于测地几何形状，事实证明，这种差异过程几乎肯定是局部恒定的。一组非构型连接到地球学的脱节，并继承了显着的分形特性。特别是，已经确定，当仅空间坐标变化时，差异概况的非构型集合具有Hausdorff dimension $ 1/2 $，并且与零集布朗尼运动的相似之处很强。这些参数至关重要的是单调性属性，当探测过程的时间结构时，这是不存在的，因此需要开发新方法。 在本文中，我们提出了几个新想法，并证明了2D差异轮廓的非构度集和1D时间过程（当空间坐标为固定和时间坐标时，分别具有hausdorff尺寸$ 5/3 $和$ 2/3 $。在我们的分析中，一个特别关键的成分是，在大地测量的“零集”上支持的，类似于布朗时代的地球时代的当地时间过程的新颖构造。此外，我们表明后者的Hausdorff Dimension $ 1/3 $与零尺寸的零集动作相比，尺寸为$ 1/2。
The Directed Landscape, a random directed metric on the plane (where the first and the second coordinates are termed spatial and temporal respectively), was constructed in the breakthrough work of Dauvergne, Ortmann, and Virág, and has since been shown to be the scaling limit of various integrable models of Last Passage percolation, a central member of the Kardar-Parisi-Zhang universality class. It exhibits several scale invariance properties making it a natural source of rich fractal behavior. Such a study was initiated in Basu-Ganguly-Hammond, where the difference profile i.e., the difference of passage times from two fixed points (say $(\pm 1,0)$), was considered. Owing to geodesic geometry, it turns out that this difference process is almost surely locally constant. The set of non-constancy is connected to disjointness of geodesics and inherits remarkable fractal properties. In particular, it has been established that when only the spatial coordinate is varied, the set of non-constancy of the difference profile has Hausdorff dimension $1/2$, and bears a rather strong resemblance to the zero set of Brownian motion. The arguments crucially rely on a monotonicity property, which is absent when the temporal structure of the process is probed, necessitating the development of new methods. In this paper, we put forth several new ideas, and show that the set of non-constancy of the 2D difference profile and the 1D temporal process (when the spatial coordinate is fixed and the temporal coordinate is varied) have Hausdorff dimensions $5/3$ and $2/3$ respectively. A particularly crucial ingredient in our analysis is the novel construction of a local time process for the geodesic akin to Brownian local time, supported on the "zero set" of the geodesic. Further, we show that the latter has Hausdorff dimension $1/3$ in contrast to the zero set of Brownian motion which has dimension $1/2.$
