Limiting Shape of First-Passage Percolation

物理学家和其他科学家使用概率的方法来模拟大量相互作用组件的动力学。例如，二维模型描述了培养皿中细菌菌落的生长或流体在随机介质中的扩散。为了更好地理解这样的系统，已经对例如在细菌菌落中，由殖民地占据的区域的边界如何随时间变化给予了很多关注。Kardar，Parisi和Zhang（KPZ）提出，可以使用随机偏微分方程来描述这些边界的演化。该项目将研究被认为受该方程支配的一大类系统，目的是验证时间演化是否被该概率模型很好地描述。该项目还为研究生提供了研究培训机会。KPZ普适性类是平面随机增长模型的集合，所有模型都表现出相同的渐近行为。在过去的二十年里，对精确可解（代数）模型的理解有了很大的进展，比如指数最后一次通过渗流和均匀随机排列中最长的递增子序列。这个项目将研究第一通道渗流。这是一类被认为属于KPZ普适类的模型，并且与上面提到的模型具有相同的渐近性质。但是，如果没有精确可解模型的代数工具，它们被证明是非常难以分析的。该项目旨在了解首过渗滤的基本特性，例如过程在不同方向上的相对增长率。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Physicists and other scientists use methods from probability to model the dynamics of large collections of interacting components. Two-dimensional models describe, for example, the growth of bacterial colonies in a petri dish or the spreading of a fluid in a random medium. To better understand such systems, much attention has been paid to how, for example in a bacterial colony, the boundary of the region occupied by the colony changes over time. Kardar, Parisi, and Zhang (KPZ) proposed that a stochastic partial differential equation could be used to describe the evolution of these boundaries. This project will study a large class of systems that are believed to be governed by this equation, with the goal of verifying that the time evolution is well-described by this probabilistic model. The project also provides research training opportunities for graduate students.The KPZ universality class is a collection of models of random growth in the plane that all exhibit the same asymptotic behavior. Over the last two decades there has been great progress understanding exactly solvable (algebraic) models like exponential last-passage percolation and the longest increasing subsequence in a uniformly random permutation. This project will study first-passage percolation. This is a class of models which are believed to be in the KPZ universality class and have the same asymptotic properties as the models mentioned above. But without the algebraic tools of the exactly solvable models, they have proven very difficult to analyze. This project aims to understand basic properties of first-passage percolation such as the relative rate of growth of the process in different directions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.