Statistical Mechanics and the Probability Theory

来自统计力学的概率模型是研究小尺度随机性如何产生全局尺度现象的框架，例如相变，它本质上是非随机的。亚历山大建议调查这个问题的以下几个方面。(1)晶格聚合物和界面由于晶格低维子空间中存在的电位而发生钉住和脱落。特别的重点将放在无序的情况下，其中电位随位置随机变化。子空间中的一个势和整体中的随机势之间的竞争也将被考虑。(2)可以从局部相互作用严格推导出溶液冻结的某些特征的模型。(3)伊辛模型以固定少数自旋的过量为条件，以显示大液滴的形成，并强调该过量的临界尺寸和超过该临界尺寸阈值时液滴形成“过渡”的宽度。(4)移动的界面，例如在Ising模型的阶段之间，可能会在减弱的相互作用降低界面能量成本的位置“挂起”，导致必须跨越的能量障碍。(5)二维波茨模型协方差矩阵的特征值及其与FK模型中某些概率衰减率的关系。(6)外场与边界条件相对立的波茨模型。就智力价值而言，这项工作是数学家和物理学家正在努力理解自然世界中各种系统的一部分，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。例子包括(i)材料的磁性；（ii）穿过不规则物质的波，如穿过地壳的地震波；（三）半导体中的杂质；（iv） DNA变性；(5)液体通过多孔物质渗透，如水或油通过地下岩石。人们早就认识到，小尺度随机性和宏观性质（包括临界现象）之间关系的许多定性方面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，而不是专门为磁铁、多孔岩石或物理世界的任何其他特定部分建模。系统只需要表现出并行特征，如集群和临界现象。亚历山大将要研究的一些系统——渗透、随机簇模型、伊辛和波茨模型以及其他自旋系统——就是这种抽象系统的例子。Alexander将研究的其他系统则更密切地基于特定的物理系统，如聚合物、DNA分子和溶液。

0405915Alexander 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) Pinning and depinning of lattice polymers and interfaces due to potentials existing in lower-dimensional subspaces of the lattice. Particular emphasis will be placed on the disordered case, in which the potential varies randomly with location. Competition between a potential in a subspace and random potentials in the bulk will also be considered. (2) Models in which certain features of the freezing of solutions can be rigorously derived from a local interaction. (3) The Ising model, conditioned, by fixing an excess of the minority spin, so as to exhibit the formation of a large droplet, with emphasis on the critical size of this excess and the width of the droplet-formation "transition" as this critical size threshold is crossed. (4) Moving interfaces, e.g. between phases of the Ising model, which may "hang up" at locations where a weakened interaction reduces the energetic cost of the interface, resulting in an energy barrier which must be crossed. (5) Eigenvalues of the covariance matrix of the two-dimensional Potts model and their relation to decay rates of certain probabilities in the FK model. (6) Potts models in which the external field(s) and the boundary condition are opposed to each other. With regard to intellectual merit, 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; (iv) denaturation of DNA; and (v) 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. Some of the systems, which Alexander will investigate--percolation, random cluster models, Ising and Potts models, and other spin systems--are examples of such abstract systems. Other systems, which Alexander will investigate, are somewhat more closely based on specific physical systems, such as polymers, DNA molecules, and solutions.