Particle Systems, Percolation, and Scaling Limits

在概率论的一般领域，近年来，在空间结构中观察到随机效应的系统发挥了越来越重要的作用，而不是(或除了)作为时间函数的行为(如在股票价格模型中)。其中一些模型，如渗流，已经单独出现在多种环境中，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。还有与组合优化(如最小生成树问题，这是设计最优路线的经典旅行商问题的变体)的联系，从而与理论计算机科学的不同部分有关。这项研究项目是关于这类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。生成树问题的一个特殊研究问题是，最优路线何时可以远离起点和终点--这一现象对于想要优化机票价格的有经验的航空旅行者来说是已知的。这项拨款下的工作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维泊松点过程的首次通过渗流，特别强调双无限测地线的不存在。第二个焦点是证明中维临界Bernoulli渗流不存在无限簇的经典问题；相关问题涉及最小生成森林中的树的数目及其对空间维度的依赖。该项目的第三部分是证明在最近建立的二维伊辛模型的近临界标度极限中关联的指数衰减。最后，第四个主题是利用标记的布朗网和网的特殊时空点来构造选民模型变体的标度极限，例如Q态Potts模型(在一维空间中)。

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 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. There are also connections to combinatorial optimization (as in the minimal spanning tree problem, which is a variant of the classical travelling salesman problem of designing optimal routings) and thus to various parts of theoretical computer science. This research project concerns the mathematical phenomena that occur in several representative examples of such stochastic systems with interesting spatial structure. One particular issue of study in the spanning tree problem is when an optimal route can go very far away from its starting and ending points -- a phenomenon known to experienced air travelers who want to optimize ticket price.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 specific topic is first-passage percolation based on d-dimensional Poisson point processes, with particular emphasis on nonexistence of doubly infinite geodesics. A second focus is on the classic problem of proving absence of an infinite cluster for intermediate dimensional critical Bernoulli percolation; a related problem concerns the number of trees in the minimal spanning forest and its dependence on spatial dimension. A third part of the project is to prove exponential decay of correlations in the recently constructed near-critical scaling limit of the two-dimensional Ising model. Finally, a fourth topic is the construction of scaling limits of voter model variants such as the q-state Potts model (in one space dimension) by using marked special space-time points of the Brownian web and net.