Studies in First-Passage Percolation and Random Walks with Scenery

9815226 Howard这项资助下的工作包括概率论一般范畴内的两个研究领域。第一个问题是在某些欧几里得第一通道渗透模型（欧几里得FPP）的背景下无限测地线的收敛速率（到相关的渐近形状）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，主要是由于与晶格效应相关的技术困难。欧几里得FPP模型发生在齐次泊松过程构造的图上，其统计性质在所有刚性运动下都具有完全不变性。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走时产生的一系列问题，其中一些固定但可能随机的着色或“风景”被分配给整数。在这种情况下，在整数上的随机行走产生沿着行走“观察到”的风景记录。被调查的问题，宽泛地说，是通过观察一个这样的风景记录可以推断出关于风景的什么。困难的根源在于人们不知道行走的步伐，只知道沿途所看到的风景。这两个领域的研究都涉及与某种类型的空间结构相关的随机过程。虽然本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在与许多有趣的物理现象相关时很自然地出现。例如，标准FPP模型最初是作为流体流过随机多孔介质（如含水层）或较小物理尺度（如膜）的表示而引入的。它还与材料科学的其他方面有联系，包括裂纹的形成和扩展。

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.