AF: Small: Markov Chains, Statistical Physics, and Mobile Geometric Graphs

AF：小：马尔可夫链、统计物理和移动几何图

• 批准号: 1016896
• 负责人: Alistair Sinclair
• 金额: $ 49.81万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016896&HistoricalAwards=false
• 关键词: AF; Small; Markov; Chains; Statistical

The project has three main themes: 1. Markov chain Monte Carlo algorithms: The study of techniques such as lifting to speed up algorithms based on Markov chains, as well as the analysis of Markov chains---for lattice triangulations and the random cluster model---that are beyond the range of current techniques.2. Statistical physics and computation: A continuing study of the Glauber dynamics for spin systems, focusing especially on the key open question of the influence of boundary conditions on the mixing time; the harnessing of an emerging understanding of spatial mixing to derive new algorithms and complexity results for counting and sampling problems; the application of new techniques for the analysis of the Boltzmann equation in physics to a computational study of nonlinear models in population genetics and genetic algorithms.3. Mobile geometric graphs: A mathematically rigorous investigation of the effects -- in terms of both increased power and novel algorithmic challenges -- of introducing mobile nodes into models of wireless networks.Numerous connections among the themes provide intellectual coherence. For example, Markov chains play a central role in statistical physics through the Glauber dynamics; phase transitions and threshold phenomena appear in all three themes, as does the pervasive notion of dynamical evolution of a system over time; and the understanding of mobile geometric graphs is intimately connected with continuum percolation in physics.In addition, all three themes are examples of outreach from theoretical computer science to other disciplines, notably probability theory, statistical physics and wireless networking, and the project is expected to contribute to cross-fertilization between computer science and these fields. Throughout the project, the choice of research questions is driven not only by their intrinsic significance but also by the challenges that they present to existing techniques and the extent to which they illuminate connections with these other fields.

该项目有三个主题： 1. 马尔可夫链蒙特卡罗算法：基于马尔可夫链的提升加速算法等技术的研究，以及马尔可夫链的分析——格子三角剖分和随机聚类模型---超出了现有技术的范围。2．统计物理和计算：对自旋系统的格劳伯动力学的持续研究，特别关注边界条件对混合时间影响的关键开放问题；利用对空间混合的新兴理解来导出计数和采样问题的新算法和复杂性结果；物理学中玻尔兹曼方程分析新技术在群体遗传学和遗传算法中非线性模型计算研究中的应用。 3．移动几何图：对将移动节点引入无线网络模型的影响（从增加的功率和新颖的算法挑战方面）进行严格的数学研究。主题之间的众多联系提供了知识的连贯性。 例如，马尔可夫链通过格劳伯动力学在统计物理学中发挥着核心作用；相变和阈值现象出现在所有三个主题中，系统随时间动态演化的普遍概念也是如此；对移动几何图的理解与物理学中的连续渗透密切相关。此外，所有三个主题都是理论计算机科学向其他学科（特别是概率论、统计物理学和无线网络）延伸的例子，该项目预计将为计算机科学和这些领域之间的交叉融合做出贡献。 在整个项目中，研究问题的选择不仅取决于其内在意义，还取决于它们对现有技术提出的挑战以及它们阐明与其他领域的联系的程度。