Probability Models from Statistical Mechanics

9802368统计力学中的亚历山大概率模型是一个框架，用于研究小规模随机性如何产生全球范围的现象，如相变，这些现象本质上是非随机的。亚历山大建议对这一主题的以下几个方面进行研究。(1)用随机团簇表示法研究Gibbs唯一性、稀释性及其对临界温度、无序模型、弱相互作用晶格气体、临界点的界限以及作为系统参数函数的各种量的单调性的影响。(2)二维系统(渗流和伊辛模型)，条件是显示出大液滴的形成，即被长轮廓包围的宏观区域，重点是该液滴的实际形状和理想形状之间的差异。(3)五彩纸屑渗流，这是一种既是各向同性的模型，又是二维的自对偶模型，因此是研究共形不变性及其相关现象的自然环境。(4)使用渗流思想来创建模拟含有杂质的水冻结的某些特征的模型。(5)某些渗流连续介质模型中有限团簇的几何形状及其与团簇大小分布的关系。这项工作是数学家和物理学家正在进行的一项努力的一部分，目的是了解自然界中的各种系统，在这些系统中，非随机的全球尺度现象反映了小规模随机性的各个方面。例如：(I)材料的磁性；(Ii)通过不规则材料的波，例如穿过地壳的地震波；(Iii)半导体中的杂质；(Iv)液体通过多孔材料的渗流，例如水或石油通过地下岩石。长期以来，人们一直认为，小规模随机性与宏观性质之间关系的许多定性方面，包括临界现象，并不太依赖于所研究的特定系统。因此，人们可以通过研究抽象系统来洞察现实世界的现象，这些抽象系统并不是专门为磁体、多孔岩石或物理世界的任何其他特定部分建模的。系统只需要展示并行特征，例如集群和临界现象。亚历山大将要研究的系统--渗流、随机团簇模型、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.