论文标题
最佳的尾巴指数在上一般的最后一段渗透中通过引导和地球几何形状
Optimal tail exponents in general last passage percolation via bootstrapping & geodesic geometry
论文作者
论文摘要
我们考虑$ \ Mathbb z^2 $的最后一段渗透，并具有一般权重分布，这预计将是Kardar-Parisi-Zhang（KPZ）通用类的成员。在此模型中，给定端点之间的面向路径最大化I.I.D的总和。与其顶点相关的重量变量称为大地测量。 Under natural conditions of curvature of the limiting geodesic weight profile and stretched exponential decay of both tails of the point-to-point weight, we use geometric arguments to upgrade the assumptions to prove optimal upper and lower tail behavior with the exponents of $3/2$ and $3$ for the weight of the geodesic from $(1,1)$ to $(r,r)$ for all large finite $r$.证据融合了几个想法，包括上一段值的众所周知的超级添加性能，拉伸指数随机变量之和的度量行为集中度以及来自地球学研究的研究和更通用的物体的几何见解。以前，这种最佳行为仅以准确的可解决模型而闻名，并依赖于对整合概率的公式进行硬分析，而公式在一般环境中不可用。我们的结果说明了一类KPZ随机生长模型中普遍性的一面，并提供了Gue Tracy-Widom分布的上和下尾指数的几何解释，该指标猜想了此类模型的一个点缩放限制。关键论点是基于对一般兴趣的观察，即超级添加允许自然迭代的引导程序获得改进的尾巴估计。
We consider last passage percolation on $\mathbb Z^2$ with general weight distributions, which is expected to be a member of the Kardar-Parisi-Zhang (KPZ) universality class. In this model, an oriented path between given endpoints which maximizes the sum of the i.i.d. weight variables associated to its vertices is called a geodesic. Under natural conditions of curvature of the limiting geodesic weight profile and stretched exponential decay of both tails of the point-to-point weight, we use geometric arguments to upgrade the assumptions to prove optimal upper and lower tail behavior with the exponents of $3/2$ and $3$ for the weight of the geodesic from $(1,1)$ to $(r,r)$ for all large finite $r$. The proofs merge several ideas, including the well known super-additivity property of last passage values, concentration of measure behavior for sums of stretched exponential random variables, and geometric insights coming from the study of geodesics and more general objects called geodesic watermelons. Previously such optimal behavior was only known for exactly solvable models, with proofs relying on hard analysis of formulas from integrable probability, which are unavailable in the general setting. Our results illustrate a facet of universality in a class of KPZ stochastic growth models and provide a geometric explanation of the upper and lower tail exponents of the GUE Tracy-Widom distribution, the conjectured one point scaling limit of such models. The key arguments are based on an observation of general interest that super-additivity allows a natural iterative bootstrapping procedure to obtain improved tail estimates.
