Topics in Percolationand Particle Models

9803267 Newman这项研究是概率论的一般领域，特别强调一些具有有趣空间结构的随机模型。一个特定的主题是基于d维泊松点过程的第一通道渗透，特别强调与半无限和双无限测地线的存在和不存在有关的结果。研究的第二个主要部分涉及随机环境中的一类选民模型，这些模型表现出有趣的“混沌时间依赖”现象。这是当模型（在时间上是一个马尔可夫过程）有多个不变分布，并且在时间t（从随机选择的初始状态开始）的分布在t趋于无穷时不收敛。一个较小但重要的研究部分涉及平面上最小生成树的连续尺度限制。在概率论的一般领域，近年来，随机效应在空间结构中而不是（或除了）作为时间函数的行为（如股票价格模型）中被观察到的系统扮演着越来越重要的角色。其中一些模型，如第一通道渗流，已经在多种情况下单独出现，例如多孔介质中的流体流动（例如与含水层中污染物分散的建模相关），聚合物结构和材料科学的其他部分。这也与组合优化有关（比如最小生成树问题，它是设计最优路线的经典旅行商问题的一个变体），因此与理论计算机科学的各个部分有关。这项资助下的研究涉及在具有有趣空间结构的随机系统的几个代表性例子中发生的数学现象。研究的一个特殊问题是空间环境中的随机性和时间上的混沌依赖性之间的联系。

9803267 Newman This research is in the general area of Probability Theory with special emphasis on a number of stochastic models with interesting spatial structure. One specific topic is first- passage percolation based on d-dimensional Poisson point processes, with particular emphasis on results related to the existence and nonexistence of semi-infinite and doubly infinite geodesics. A second major part of the work concerns a class of voter models in random environments which exhibit the intriguing phenomenon of "chaotic time dependence." This is when the model (which is a Markov process in time) has multiple invariant distributions and the distribution at time t (starting from a randomly selected initial state) does not converge as t tends to infinity. A smaller, but significant, part of the research involves continuum scaling limits of minimal spanning trees in the plane. In the general area of probability theory, an increasingly important role has been played in recent years by systems in which random effects are observed in the spatial structure rather than in (or in addition to) the behavior as a function of time (as in models of equity prices). Some of these models, such as first-passage percolation, have arisen separately in multiple contexts, such as fluid flow in porous media (which is relevant for example to modeling of pollutant dispersion in aquifers), polymer structure, and other parts of materials science. There are also connections to combinatorial optimization (as in the minimal spanning tree problem, which is a variant of the classical traveling salesman problem of designing optimal routings) and thus to various parts of theoretical computer science. The research under this grant concerns the mathematical phenomena that occur in several representative examples of such stochastic systems with interesting spatial structure. One particular issue of study is the connection between randomness in the spatial environment and chaotic depe ndence in time.