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

该项目有三个主要主题：1.马尔可夫链蒙特卡罗算法：研究基于马尔可夫链的提升加速算法等技术，以及分析超出当前技术范围的马尔可夫链-用于格子三角剖分和随机簇模型。统计物理和计算：继续研究自旋系统的Glauber动力学，特别关注边界条件对混合时间的影响这一关键的公开问题；利用对空间混合的新兴理解来推导计数和采样问题的新算法和复杂性结果；将物理中分析Boltzmann方程的新技术应用于群体遗传学和遗传算法中的非线性模型的计算研究。移动几何图：对将移动节点引入无线网络模型的影响进行了严格的数学研究--在功率增加和新的算法挑战方面。主题之间的大量联系提供了智力上的一致性。例如，马尔可夫链通过Glauber动力学在统计物理中发挥核心作用；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中的连续统渗透密切相关。此外，所有这三个主题都是理论计算机科学扩展到其他学科的例子，特别是概率论、统计物理和无线网络，该项目有望促进计算机科学和这些领域之间的相互促进。在整个项目中，选择研究问题不仅是因为它们的内在意义，也是因为它们对现有技术提出的挑战，以及它们在多大程度上阐明了与这些其他领域的联系。

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.