Particle Systems, Percolation, and Scaling Limits

近年来，在概率论的一般领域，随机效应是在空间结构中观察到的，而不是在作为时间函数的行为中（或除了作为时间函数的行为之外）（如股票价格模型），这些系统发挥了越来越重要的作用。其中一些模型，如渗流，分别出现在多种情况下，如多孔介质中的流体流动（例如与含水层中污染物扩散的建模有关），聚合物结构和材料科学的其他部分。它还与组合优化（如最小生成树问题，这是设计最优路径的经典旅行商问题的一个变体）有关，因此与理论计算机科学的各个部分有关。本研究项目涉及的数学现象，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。在生成树问题的研究中的一个特殊问题是，当一个最佳路线可以去非常远离它的起点和终点-一个有经验的航空旅客谁想要优化票价已知的现象。在此授权下的工作是在概率论的一般领域，特别强调了一些有趣的空间结构的随机模型。一个具体的主题是基于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性。第二个重点是经典的问题，证明没有一个无限集群的中间维临界伯努利渗流;一个相关的问题涉及的树木的数量在最小生成森林和它的依赖空间维度。该项目的第三部分是证明指数衰减的相关性在最近构建的近临界尺度限制的二维伊辛模型。最后，第四个主题是建设的缩放限制的选民模型的变种，如q-状态波茨模型（在一个空间维度）通过使用标记的特殊时空点的布朗网络和网络。

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.