Studies in First-Passage Percolation and Random Walks with Scenery

9815226霍华德 这项拨款的工作包括两个调查领域， 概率论的范畴。 第一个问题涉及收敛速度（到相关的 渐近形状）和有关性质的无限测地线在某些背景下， 欧几里德第一通道渗流模型（Euclidean FPP）。严格无条件 在标准FPP的上下文中，关于无限测地线（时间最小化路径）的结果是 相当稀疏，主要是由于与晶格效应相关的技术困难。 模型 欧氏FPP发生在由齐次泊松过程构造的图上 它们的统计性质在所有刚体运动下都具有完全的不变性。 在这 从某种意义上说，欧几里德模型是研究FPP测地线的自然环境。 的 第二个领域涉及整数上的随机游动引起的大量问题，其中 某些固定但可能随机的着色或"风景"被分配给整数。 在这 在上下文中，整数上的随机游走产生沿着沿着 步行. 粗略地说，调查中的问题涉及可以推断出什么 通过观察这样的风景记录来了解风景。 困难来自于 事实上，一个人不知道步行所采取的步骤，只有沿着观察到的风景 散步 这两个领域的调查涉及随机过程与某种类型 空间结构。 虽然这项研究的重点是数学方面的这些 模型，这种一般类型的模型很自然地出现在与一些 有趣的物理现象。 例如，标准FPP模型是第一个 作为通过随机多孔介质的流体流动的表示引入， 含水层，或在较小的物理尺度上，膜。 也有连接到其他 材料科学的各个方面，包括裂纹的形成和传播。

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.