Topics in Percolation and Particle Models

在这项资助下的工作是在概率论的一般领域，特别强调一些具有有趣空间结构的随机模型。C.D.Howard的一个项目涉及欧几里德首次通过渗流，目的是证明双无限测地线的不存在，并导出迄今仅用精确解方法证明的具有特殊组合结构的模型的二维涨落指数的值。其他合作项目是关于交互粒子系统的。一个是L.R.Fontes，M.Isopi和K.Ravishankar，他们关注的是年龄、比例限制和随机利率选民模型等系统中的混沌时间依赖性。另一类是与F.Camia，E.de Santis等人研究零温随机Ising模型的局部瞬时性、常返性和吸引性问题，包括具有随机环境的模型。在概率论的一般领域中，随机效应在空间结构中而不是在作为时间函数的行为(如在股票价格模型中)中观察到的系统扮演着越来越重要的角色。其中一些模型，如首次通过渗流，已经单独出现在多个背景下，如多孔介质中的流体流动(例如与含水层中污染物扩散的模拟有关)、聚合物结构和材料科学的其他部分。这项拨款下的研究涉及这种具有有趣空间结构的随机系统的几个典型例子中出现的数学现象。虽然主要关注的是简化模型的严格数学结果，但长期目标是提供新的理解和工具，这将对应用中使用的更复杂的模型有用。

The work under this grant is in the general area of Probability Theory with special emphasis on a number of stochastic models with interesting spatial structure. One project, with C.D. Howard, concerns Euclidean first-passage percolation and aims to prove the nonexistence of doubly infinite geodesics and derive the values of two-dimensional fluctuation exponents that to date have only been proved using exact solution methods for models with special combinatorial structure. Other collaborative projects are on interacting particle systems. One is with L.R. Fontes, M. Isopi and K. Ravishankar and concerns aging, scaling limits and chaotic time dependence in such systems as voter models with random rates. Another is with F. Camia, E. De Santis and others and concerns local transience, recurrence and absorption issues for zero-temperature stochastic Ising models, including those with random environments.In the general area of probability theory, an increasingly important role is played 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 modelling of pollutant dispersion in aquifers), polymer structure and other parts of materials science. The research under this grant concerns the mathematical phenomena that occur in several representative examples of such stochastic systems with interesting spatial structure. Although the main focus is on rigorous mathematical results for simplified models, the long term goal is to provide new understanding and tools that will be useful for the more complex models used in applications.