Mathematical Sciences: Limit Theorems in Statistical Mechanical Setting

数学科学：统计力学环境中的极限定理

• 批准号: 9504513
• 负责人: Dmitry Ioffe
• 金额: $ 4万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504513&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Limit; Theorems; Statistical

9504513 Ioffe Abstract The research is devoted to a study of various limit phenomena which arise in a statistical mechanical context or are closely related to it. More specifically, the investigator works - partly in a collaboration with other researchers - on questions concerning the structure of limit theorems in the phase transition regime as well as on applications of ideas reminiscent of the concept of complete analyticity to a general probabilistic setting such as, for example, to the theory of Markov chains or to the theory of random fields. Principal mathematical issues addressed in this research include further investigation of the two dimensional Ising model in the whole of the phase transition region, analysis of Ornstein-Zernike behavior of two-and-higher dimensional correlations and related shape and fluctuation theorems, analysis of limit phenomena for almost Markovian random fields and justification of the Wulff construction in various two and three dimensional models (e.g. SOS, Ising and continuous spin Ising, percolation etc.), based on sharp subvolume order large deviation results. Long term objectives of this research are twofold. On the one hand, powerful ideas and methods developed over the past decade in mathematical statistical mechanics--such as, for example, the method of polymer expansions, the concept of complete analyticity and the method of induction in volume--may provide both useful insights and approaches to various limit problems in probability and stochastic processes. On the other hand, most of the problems addressed in this research emerge from physical and material science theories of a rigorous reconstruction of thermodynamical and kinetic properties of matter directly from microscopic considerations. Thus, a successful investigation is ultimately aimed at a better understanding of such issues as, for example, equilibrium structure and stability of crystals and dynamics of surface growth.

9504513 Ioffe 摘要 该研究致力于研究统计力学背景下出现的或与之密切相关的各种极限现象。更具体地说，研究者部分地与其他研究人员合作，研究有关相变体系中极限定理结构的问题，以及将完全解析性概念的思想应用于一般概率环境，例如马尔可夫链理论或随机场理论。本研究解决的主要数学问题包括进一步研究整个相变区域的二维伊辛模型、分析二维及更高维相关性的奥恩斯坦-泽尼克行为以及相关的形状和波动定理、分析几乎马尔可夫随机场的极限现象以及各种二维和三维模型（例如SOS、伊辛和连续自旋伊辛、 渗流等），基于尖锐的子体积顺序大偏差结果。 这项研究的长期目标是双重的。一方面，过去十年在数理统计力学中发展起来的强有力的思想和方法——例如聚合物展开的方法、完全解析性的概念和体积归纳法——可能为概率和随机过程中的各种极限问题提供有用的见解和方法。另一方面，本研究解决的大多数问题都来自物理和材料科学理论，直接从微观角度严格重建物质的热力学和动力学性质。因此，成功的研究最终旨在更好地理解晶体的平衡结构和稳定性以及表面生长动力学等问题。