Statistical Mechanics and the Probability Theory

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，而相变本质上是非随机的。亚历山大建议对这一主题的以下几个方面进行研究。(1)由于晶格的低维子空间中存在势，晶格聚合物和界面的钉扎和去钉扎。我们将特别强调无序情况，在这种情况下，电势随位置随机变化。子空间中的势和整体中的随机势之间的竞争也将被考虑。(2)溶液冻结的某些特征可以从局部相互作用中严格推导出来的模型。(3)Ising模型，通过固定少数自旋的过剩来显示大液滴的形成，重点是这种过剩的临界尺寸和当超过这一临界尺寸阈值时液滴形成“转变”的宽度。(4)移动的界面，例如在伊辛模型的两个阶段之间，这可能会在相互作用减弱降低界面能量成本的位置“挂起”，从而导致必须跨越的能量屏障。(5)二维Potts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(S)和边界条件相互对立的Potts模型。关于智力价值，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。这些例子包括：(1)材料的磁性；(2)波在不规则材料中传播，如地震波穿过地壳；(3)半导体中的杂质；(4)DNA的变性；(5)液体通过多孔材料的渗透，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。亚历山大将研究的一些系统--渗流、随机星团模型、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。亚历山大将研究的其他系统在某种程度上更紧密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.