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

这里的项目研究空间随机模型，特别是渗透和短程（现实）自旋玻璃。主要问题涉及第一次通过渗流，独立的渗流模型在临界阈值，和地面（零温度）的爱德华-安德森自旋玻璃模型的状态。从分析的角度来看，一些项目试图了解相关随机变量集合的最大值或最小值的结构和波动;这不仅适用于测地线和首次通过渗流方差估计的问题，而且适用于自旋玻璃态的组织。另一方面，这些技术有几何的味道，包括遍历理论和伯努利渗透的工具。预计所获得的结果将影响其他数学领域，例如聚合物模型，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，通过多孔介质的流体流动，随机增长模型和稀金属合金的磁性。这里研究的项目集中在这些模型上，并关注定义随机环境的小尺度规则与系统的大尺度属性之间的联系。虽然其中一些属性高度依赖于局部规则的性质，但其他属性是“普遍的”，并且似乎只依赖于周围空间的维度。这个项目的一个主要目标是了解这种普遍行为的起源。结果不仅对相关的数学领域，而且对物理学，计算机科学，生物学和金融数学等当前感兴趣的主题都有影响。

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.