Statistical Mechanics and Related Probability Theory

统计力学中的概率模型是研究小尺度随机性如何产生全局尺度现象（如相变）的框架，而相变本质上是非随机的。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)三维Ising模型中的分层过渡，在没有外场的情况下，仅由界面对壁的吸引力和熵排斥竞争引起。(3)收敛到平衡，在模拟溶液冻结时发生的相分离的系统中。(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。该研究有助于为聚合物、粘附和超导等领域的应用工作提供理论基础。这项工作可能会导致可以被实验家验证的预测，从而导致新的实际发现。将数学家的观点引入到以前主要由物理学家考虑的理论问题中，通常会改变物理学家的观点，从而导致进一步的进步。亚历山大与Francois Dunlop和Salvador Miracle-Sole的合作加强了美法scientiÞc合作关系。这项工作是数学家和物理学家正在努力理解自然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尺度随机性的各个方面。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、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将要研究的其他系统在某种程度上更密切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.