Probabilistic Analysis of Random Geometric Structures

本文主要研究欧几里得空间中随机结构的严格概率分析。在前人研究的基础上，我们着重于(i)二维或多维随机填充、随机生长界面和松散球体填充的随机分析，以及（ii）计算几何和组合优化中的布尔模型和随机欧几里得点集的泛函。(i)的目的是建立热力学极限、高斯波动和泛函中心极限定理，这些泛函与填充模型、界面模型以及一般的随机欧几里德点集有关。所有这些模型都表现出有限范围的相互作用和长期依赖。第二部分着重于欧几里德空间中随机点集泛函的概率分析。在（ii）中提出的工作将进一步加深我们对欧几里得优化问题、计算几何中的随机图和随机数据的统计分析的解决方案的理解。物理、化学和生物过程的中心问题之一包括理解随机界面的生长。这些界面在理解化学沉积、维管网络形态、裂缝分析中的生长模式和介质孔隙度时自然出现。随时间变化的动态随机界面，对于理解和模拟晶体生长以及理解自然界随时间变化的模式非常有用。这包括各种各样的现象，如疾病在人口中的传播、森林火灾中的生长模式，以及对飞机上发现的金属的断裂分析。我们对随机界面的理解大多只是实验性的。这一建议旨在将我们对随机界面的理解建立在坚实的理论基础上。了解随机界面的行为目前是首席研究员与利哈伊大学工程学院的科学家以及朗讯科技公司的科学家合作的核心部分。

0203720Yukich This proposal focuses on the rigorous probabilistic analysis of random structures in Euclidean space. Building on previous research, we focus on the stochastic analysis of (i) random packing, random growing interfaces, and loose sphere packing, all in two or more dimensions, and (ii) functionals of Boolean models and random Euclidean point sets in computational geometry and combinatorial optimization. The aim in (i) is to establish thermodynamic limits, Gaussian fluctuations, and functional central limit theorems for functionals associated with packing models, interface models, as well as for random Euclidean point sets in general. All of these models exhibit finite range interactions and long-range dependence. Part (ii) focuses on the probabilistic analysis of functionals of random point sets in Euclidean space. The proposed work in (ii) will further our understanding of solutions to Euclidean optimization problems, random graphs in computational geometry, and the statistical analysis of random data. One of the central problems in physical, chemical, and biological processes involves understanding the growth of random interfaces. Such interfaces arise naturally in understanding chemical deposition, the morphology of vascular networks, growth patterns in fracture analysis, and the porosity of media. Dynamic random interfaces, ones which change with time, are useful in understanding and modeling crystal growth as well as understanding patterns in nature which change with time. This includes diverse phenomena such as the spread of disease in a population, the growth patterns in a forest fire, as well as the fracture analysis of metals such as those found in aircraft. Much of our understanding of random interfaces is only experimental. This proposal seeks to put our understanding of random interfaces on a solid theoretical foundation. Understanding the behavior of random interfaces currently forms a central part of the principal investigator's collaboration with scientists from the Lehigh University engineering school as well as with scientists from Lucent Technologies.