Statistical Mechanics and Related Probability Theory

统计力学中的概率模型是研究小规模随机性如何产生全球范围的现象的框架，例如相变，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)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)三维伊辛模型中无外场作用下的分层相变，它完全是由界面与壁的相互吸引与熵排斥作用引起的。(3)收敛到平衡，在对溶液冻结时发生的相分离进行建模的系统中。(4)移动的界面，例如在Ising模型的相间，它可能“挂起”在减弱的相互作用降低界面的能量成本的位置，导致必须跨越的能量变化。(5)二维Potts模型的协方差矩阵的本征值及其与FK模型中某些概率的衰减率的关系。(6)外场(S)和边界条件相互对立的POTS模型。这一研究有助于为聚合物、粘合和超导等领域的应用工作提供理论支持。这项工作可能会带来可以被实验者检验的预测，从而带来新的实际发现。将数学家的观点引入以前主要由物理学家考虑的理论问题，通常可能会改变物理学家的观点，导致进一步的进步。亚历山大与弗朗索瓦·邓洛普和萨尔瓦多奇迹-索尔的合作加强了美法科学的合作伙伴关系。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尺度随机性的各个方面。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，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、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。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。

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) Layering transitions in the three-dimensional Ising model without an external field, caused solely by the attraction of an interface to a wall in competition with entropic repulsion. (3) Convergence to equilibrium, in systems modeling the phase separation that occurs in the freezing of solutions. (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 barier 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.The research can help provide a theoretical underpinning for applied work in areas such as polymers, adhesion, and superconductivity. The work may lead to predictions which can be tested by experimentalists, leading to new practical discoveries. Introducing a mathematician's perspective to theoretical problems that have previously been considered principally by physicists may in general alter the physicists' perspective,leading to further advances. US-French scientiÞc partnership is enhanced by Alexander's collaboration with Francois Dunlop and Salvador Miracle-Sole. 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 reþect 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-scall 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, 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.