Random spatial systems and ground states of short-range spin glasses

这里的项目研究空间随机模型，特别是渗流和短程(真实)自旋玻璃。主要问题涉及Edwards-Anderson自旋玻璃模型的首次通过渗流、临界点的独立渗流模型和基态(零温度)。从分析的角度来看，一些项目试图了解一组相关随机变量的最大值或最小值的结构和波动；这不仅适用于测地线问题和首次通过渗流中的方差估计问题，也适用于自旋玻璃态的组织。另一方面，这些技术具有几何色彩，包括遍历理论和伯努利渗流的工具。预计所获得的结果将影响其他数学领域，如聚合物模型、粒子系统和理论计算机科学中的问题。随机空间系统的理论已经发展到包括随机距离模型、流体在多孔介质中的流动、随机生长模型和稀金属合金的磁性。这里所研究的项目以这样的模型为中心，关注定义随机环境的小尺度规则和系统的大尺度属性之间的联系。虽然其中一些性质高度依赖于当地规则的性质，但其他性质是“普遍的”，似乎只取决于周围空间的大小。这个项目的一个主要目标是了解这种普遍行为的起源。这些结果不仅应该对相关的数学领域产生影响，而且应该对物理、计算机科学、生物学和金融数学等当前感兴趣的课题产生影响。

The projects herein study spatial stochastic models, specifically percolation and short-range (realistic) spin glasses. The main questions concern first-passage percolation, independent percolation models at the critical threshold, and ground (zero-temperature) states of the Edwards-Anderson spin glass model. Viewed analytically, some of the projects seek to understand the structure and fluctuations of the maximum, or minimum, of a collection of correlated random variables; this applies not only to questions of geodesics and estimation of variance in first-passage percolation, but also to the organization of spin glass states. On the other hand the techniques have a geometric flavor, including tools from ergodic theory and Bernoulli percolation. It is expected that the results obtained will influence other mathematical areas, for instance polymer models, particle systems and questions in theoretical computer science.The theory of random spatial systems has grown to include models of random distances, fluid flow through porous media, random growth models and magnetic properties of dilute metallic alloys. The projects studied here center on such models and concern the connection between the small-scale rules that define the random environment and the large-scale properties of the system. While some of these properties are highly dependent on the nature of the local rules, others are ``universal'' and seem to only depend on the dimension of the ambient space. A main goal of this project is to understand the origin of this universal behavior. The results should have consequences not only for related mathematical areas, but for topics of current interest in physics, computer science, biology and financial mathematics.