Probability Models from Statistical Mechanics

9802368亚历山大 统计力学的概率模型是研究 小尺度的随机性会产生全球尺度的现象，例如相变， 它们本质上是非随机的。 亚历山大建议调查以下方面 的主题。 (1)使用随机簇表示来研究吉布斯唯一性， 稀释及其对临界温度的影响，无序模型，晶格气体， 弱相互作用，临界点的界限，以及各种量的单调性， 系统参数的函数。 (2)二维系统（渗流和伊辛方程） 模型）调节以显示大液滴的形成，即宏观的 由长轮廓包围的区域，重点是实际轮廓之间的差异 和理想的形状。 (3)五彩纸屑渗流，一个各向同性的 在两个维度上，自我对偶，因此是一个自然的环境， 研究共形不变性及相关现象。 (4)利用渗滤思想， 创建一个模型，它模仿水的冻结的某些功能，其中包含 杂质 (5)有限团簇的几何结构及其与团簇大小的关系 分布，对于某些连续渗流模型。 这项工作是数学家和物理学家正在努力的一部分， 了解自然世界中的各种系统，其中非随机的全球规模 现象反映了小规模随机性的方面。 例如：（1）磁性 材料的性质;（ii）通过不规则材料传播的波，例如地震波 波通过地壳;（iii）半导体中的杂质;（iv）渗透 液体通过多孔材料，如水或石油通过地下岩石。 它有 长期以来，人们一直认为，小尺度随机性和宏观性质之间关系的许多定性方面，包括临界现象， 不要太依赖于所研究的特定系统。 因此，可以获得 通过研究不打算建模的抽象系统来洞察现实世界的现象 特别是磁铁，多孔岩石，或者物理世界的任何其他特定部分。 系统只需要表现出并行的功能，如集群和临界现象。 亚历山大将要研究的系统--渗流，随机簇模型， 伊辛和波茨模型，以及其他自旋系统，都是这种抽象系统的例子。

9802368 Alexander Probability models from statistical mechanics are a framework for studying how small-scale randomness produces global-scale phenomena, such as phase transitions, which are essentially nonrandom. Alexander proposes to investigate the following aspects of the subject. (1) The use of random-cluster representations to study Gibbs uniqueness, dilution and its effect on critical temperatures, disordered models, lattice gases with weak interactions, bounds on critical points, and monotonicity of various quantities as a function of system parameters. (2) Two-dimensional systems (percolation and the Ising model) conditioned so as to exhibit the formation of a large droplet, that is, a macroscopic region enclosed by a long contour, with emphasis on the discrepancy between the actual and ideal shape of this droplet. (3) Confetti percolation, a model which is both isotropic and, in two dimensions, self-dual, and hence is a natural setting in which to study conformal invariance and related phenomena. (4) The use of percolation ideas to create a model which mimics certain features of the freezing of water which contains impurities. (5) The geometry of finite clusters, and its relation to the cluster size distribution, for certain continuum models of percolation. This work is part of an ongoing effort by mathematicians and physicists to understand various systems in the natural world in which nonrandom global-scale phenomena reflect aspects of small-scale randomness. Examples include (i) magnetic properties of materials; (ii) waves traveling through irregular materials, such as seismic waves through the earth's crust; (iii) impurities in semiconductors; and (iv) percolation of liquid through a porous material, such as water or oil through underground rock. It has long been understood that many qualitative aspects of the relation between small- scale randomness and macroscopic properties, including critical phenomena, do not depend too closely on the particular system being studied. One can therefore gain insight into real-world phenomena by studying abstract systems not intended to model specifically magnets, or porous rock, or any other particular part of the physical world. The systems need only exhibit parallel features, such as clustering and critical phenomena. The systems which Alexander will investigate--percolation, random cluster models, Ising and Potts models, and other spin systems--are examples of such abstract systems.