论文标题
离散的测量本地时间在KPZ缩放下收敛
Discrete geodesic local time converges under KPZ scaling
论文作者
论文摘要
（Dauvergne-Ortmann-Virag '18）中构建的定向景观产生了定向的，平面，随机的几何形状，被认为是Kardar-Paris-Paris-Paris-Parisi-Zhang（KPZ）Universital类中的二维第一和最后一段渗透模型的通用缩放限制。在这种随机几何形状中的大地学形成了一种重要的随机连续曲线，表现出波动理论与布朗运动完全不同。在这种脉络中，与布朗尼当地时间的对应物，BLT（BLT支持了布朗运动的零相似措施），最近构建了GLT的当地时间，并用于研究（Ganguly-Zhang '22）中有向景观的分形特性。这是一个经典的事实，可以使用布朗运动的马尔可夫特性来证明，该特性在简单随机行走的零件集合中均匀的离散度量收敛到BLT。在本文中，我们通过证明在限制前的可限制的最后一段渗透模型中的当地时代的当地时代通过在以KPZ指数为指导的适当缩放下收敛到GLT，从而证明了这一点的“ kpz模拟”。 在没有任何马克维亚语的情况下，我们的论点依赖于在限制模型中最近经过证实的地球化学融合在有名景观中（Dauvergne-Virag '21）。但是，该输入涉及宏观特性，并且太粗糙，无法捕获当地时间分析所需的微观信息。为了关联宏观和显微镜行为，关键成分是离散模型中当地时间的先验平滑度估计，证明是依靠几何思想，例如地球学的合并以及它们在边界数据扰动下的稳定性。
The directed landscape constructed in (Dauvergne-Ortmann-Virag '18) produces a directed, planar, random geometry, and is believed to be the universal scaling limit of two-dimensional first and last passage percolation models in the Kardar-Parisi-Zhang (KPZ) universality class. Geodesics in this random geometry form an important class of random continuous curves exhibiting fluctuation theory quite different from that of Brownian motion. In this vein, counterpart to Brownian local time, BLT (a self-similar measure supported on the set of zeros of Brownian motion), a local time for geodesics, GLT, was recently constructed and used to study fractal properties of the directed landscape in (Ganguly-Zhang '22). It is a classical fact and can be proven using the Markovian property of Brownian motion that the uniform discrete measure on the set of zeros of the simple random walk converges to BLT. In this paper, we prove the ''KPZ analog'' of this by showing that the local times for discrete geodesics in pre-limiting integrable last passage percolation models converge to GLT under suitable scaling guided by KPZ exponents. In absence of any Markovianity, our arguments rely on the recently proven convergence of geodesics in pre-limiting models to that in the directed landscape (Dauvergne-Virag '21). However, this input concerns macroscopic properties and is too coarse to capture the microscopic information required for local time analysis. To relate the macroscopic and microscopic behavior, a key ingredient is an a priori smoothness estimate of the local time in the discrete model, proved relying on geometric ideas such as the coalescence of geodesics as well as their stability under perturbations of boundary data.
