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摘要这项研究致力于对统计机械背景下出现或与之紧密相关的各种极限现象的研究。更具体地说，研究人员在与其他研究人员的合作中进行了有关相关性限制定理的结构的问题，以及在阶段过渡制度中的结构以及思想的应用，让人联想到对一般概率环境的完全分析性的概念，例如，例如，对马尔可夫链的理论或随机领域的理论理论。 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,渗透等），基于尖锐的子体积级较大的偏差结果。 这项研究的长期目标是双重的。一方面，在过去十年中，在数学统计力学中发展了强大的思想和方法 - 例如，聚合物扩展的方法，完整分析性的概念以及卷的诱导方法 - 可能提供了有用的见解和概率和随机过程的各种限制问题的有用见解和方法。另一方面，这项研究中解决的大多数问题来自物理和材料科学理论，这些理论是直接从显微镜考虑的严格重建物质和动力学特性的重建。因此，成功的研究最终旨在更好地理解此类问题，例如，晶体的平衡结构和稳定性以及表面生长的动力学。