Mathematical Sciences: Limit Theorems in Statistical Mechanical Setting

9504513约菲摘要 该研究致力于研究在统计力学背景下出现或与之密切相关的各种极限现象。更具体地说，调查工作-部分在与其他研究人员合作-对有关的问题的结构极限定理在相变制度以及对应用程序的想法让人想起的概念完全解析一般的概率设置，如，例如马尔可夫链理论或随机场理论。本研究的主要数学问题包括：进一步研究二维Ising模型在整个相变区的性质，分析二维和高维关联的Ornstein-Zernike行为及相关的形状和涨落定理，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中Wulff构造的合理性（例如SOS、Ising和连续自旋Ising、渗滤等），基于锐子体积阶差大的结果。 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，聚合物展开法、完全解析性的概念和体积归纳法-可以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，本研究中解决的大多数问题都来自物理和材料科学理论，这些理论直接从微观考虑出发，对物质的化学和动力学性质进行了严格的重建。因此，一个成功的研究最终是为了更好地了解这样的问题，例如，平衡结构和稳定性的晶体和表面生长的动力学。

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.