Studies in First-Passage Percolation and Random Walks with Scenery

第一通道渗透和风景随机行走研究

• 批准号: 9815226
• 负责人: C. Douglas Howard
• 金额: $ 7.74万
• 依托单位: CUNY Baruch College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-09-01 至 2001-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9815226&HistoricalAwards=false
• 关键词: Studies; First; Passage; Percolation; Random

9815226 Howard The work under this grant comprises two areas of investigation in the general category of probability theory. The first concerns rates of convergence (to an associated asymptotic shape) and related properties of infinite geodesics in the context of certain Euclidean first-passage percolation models (Euclidean FPP). Rigorous unconditional results about infinite geodesics (time-minimizing paths) in the context of standard FPP are quite sparse, largely due to technical difficulties associated with lattice effects. Models of Euclidean FPP take place on graphs constructed from a homogeneous Poisson process and their statistical properties enjoy complete invariance under all rigid motions. In this sense, Euclidean models are a natural setting in which to study FPP geodesics. The second area concerns a host of problems arising from random walks on the integers where some fixed but possibly random coloring, or "scenery", is assigned to the integers. In this context, a random walk on the integers produces a record of scenery "observed" along the walk. The issues under investigation concern, loosely speaking, what can be inferred about the scenery by observing one such scenery record. The difficulty stems from the fact that one does not know the steps taken by the walk, only the scenery observed along the walk. Both areas of investigation involve stochastic processes associated with some type of spatial structure. While the focus of this research is the mathematical aspects of these models, models of this general sort arise quite naturally in connection with a number of interesting physical phenomena. For example, the standard FPP model was first introduced as a representation of fluid flow through a random porous medium such as an aquifer or, on a smaller physical scale, a membrane. There are also connections to other aspects of materials science, including the formation and propagation of cracks.

9815226霍华德，这项赠款下的工作包括概率论一般范畴的两个研究领域。第一个问题是在某些欧氏首次通过渗流模型(欧几里德FPP)的背景下无限测地线的收敛速度(到一个相关的渐近形状)和相关性质。在标准FPP的背景下，关于无限测地线(时间最小化路径)的严格无条件结果相当稀疏，这主要是由于与格子效应相关的技术困难。欧氏FPP模型发生在由齐次泊松过程构成的图上，其统计性质在所有刚性运动下都是完全不变的。从这个意义上说，欧几里得模型是研究FPP测地线的自然环境。第二个领域涉及在整数上随机游动引起的一系列问题，其中一些固定的但可能是随机的着色或“风景”被分配给整数。在这种情况下，在整数上的随机漫步产生沿该漫步所观察到的风景的记录。调查中的问题，粗略地说，是指通过观察一个这样的风景记录可以推断出什么风景。困难源于这样一个事实，一个人不知道步行所走的步骤，只知道步行沿途观察到的风景。这两个研究领域都涉及与某种类型的空间结构相关的随机过程。虽然这项研究的重点是这些模型的数学方面，但这种一般类型的模型很自然地与许多有趣的物理现象有关。例如，标准的FPP模型最初是作为流体通过随机多孔介质的表示，例如含水层，或者在较小的物理尺度上，通过膜。它还与材料科学的其他方面有联系，包括裂纹的形成和传播。