Mathematical Sciences: Limit Theorems in Statistical Mechanical Setting

[504513]摘要本研究致力于研究在统计力学背景下产生的或与之密切相关的各种极限现象。更具体地说，研究者工作-部分是与其他研究人员合作-关于相变制度中极限定理结构的问题，以及在一般概率设置中使人想起完全分析概念的想法的应用，例如，在马尔可夫链理论或随飞机理论中。本研究的主要数学问题包括进一步研究整个相变区域的二维Ising模型，分析二维和高维相关的Ornstein-Zernike行为及其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，以及证明各种二维和三维模型（如SOS， Ising和连续自旋Ising）中的Wulff结构，渗流等)，基于尖锐的次容积阶偏差较大的结果。这项研究的长期目标是双重的。一方面，过去十年数理统计力学中发展起来的强大的思想和方法——例如，聚合物膨胀的方法、完全分析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来自物理和材料科学理论，这些理论直接从微观考虑对物质的热力学和动力学性质进行了严格的重建。因此，成功的研究最终是为了更好地理解诸如晶体的平衡结构和稳定性以及表面生长动力学等问题。

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.