Planar First Passage Percolation

概率论已被证明是描述统计物理学中出现的许多系统的有用框架。这些系统往往过于复杂，无法用任何计算机完全模拟。虽然精确的计算是不可能的，但数学给了我们一种了解这些系统的一些重要性质的方法。这项研究建立在物理学中一种被称为第一通道渗透的模型的最新进展之上。除了进行数学研究外，该奖项还将通过支持研究生和本科教育来帮助发展国家的科学和技术基础设施。此外，它还将赞助几场讲座，帮助向更广泛的社区解释数学研究的应用。本研究考虑了平面上首通道渗流的大量问题。近年来，利用Busemann函数对首道渗流测地线的研究取得了很大进展。这项研究将以这些结果为基础，试图回答这个领域几十年来一直存在的核心问题。这些问题包括证明独立第一通道渗流的极限形状是严格凸的，以及在任何固定方向上不存在双无限测地线。该项目将通过支持研究生和本科教育，帮助发展国家的科技基础设施。

Probability theory has proven to be a useful framework for describing many systems that arise in statistical physics. These systems are often too complicated to model completely with any computer. Although exact computations are not possible, mathematics gives us a way to learn about some important properties of these systems. This research builds on recent advances in a model from physics known as first passage percolation. In addition to conducting mathematical research this award will help to develop the nation's scientific and technological infrastructure by supporting graduate and undergraduate education. Additionally it will sponsor several talks to help explain applications of mathematical research to a broader community.This research considers a large number of questions in first-passage percolation on the plane. Recently there have been great advances in the study of geodesics in first-passage percolation using Busemann functions. This research will build on these results to try to answer central questions in the field that have been open for decades. These questions include showing that the limiting shape in independent first-passage percolation is strictly convex and that there are no bi-infinite geodesics in any fixed direction. This project will help develop the nation's scientific and technological infrastructure by supporting graduate and undergraduate education.