Topics in Percolation and Particle Models

根据这项补助金的工作是在概率论的一般领域，特别强调一些有趣的空间结构的随机模型。 一个项目，与C.D.霍华德，关注欧几里德第一次通过渗流和目的是证明不存在的双重无限测地线和推导的二维波动指数的值，到目前为止，只有证明了使用精确解方法的模型与特殊的组合结构。 其他合作项目是关于相互作用的粒子系统。 一个是L. R。丰特斯湾Isopi和K. Ravishankar和关注老化，缩放限制和混沌的时间依赖性等系统的选民模型与随机率。 另一个是F。Camia，E.在概率论的一般领域中，随机效应在空间结构中而不是在时间的函数中（如在股票价格模型中）被观察到的系统起着越来越重要的作用。 其中一些模型，如第一次通过渗流，是在多种情况下分别出现的，如多孔介质中的流体流动（例如，这与含水层中污染物扩散的建模有关）、聚合物结构和材料科学的其他部分。 根据这项补助金的研究涉及的数学现象，发生在几个有代表性的例子，这样的随机系统与有趣的空间结构。 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，这将有助于应用程序中使用的更复杂的模型。

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.