Probability and Statistical Mechanics

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，本质上是非随机的。 亚历山大建议调查以下方面的问题。 (1)伊辛模型的条件，通过固定过量的少数自旋，以便表现出一个大的液滴的形成，重点是这个过量的临界尺寸和这个液滴的实际和理想形状之间的差异。 (2)渗滤模型也有条件表现出一个大的液滴的形成，强调边界的局部波动的下限。 (3)自旋系统和渗流模型，特别是有限体积中的混合性质和相关性与连通性的衰减。 (4)低温下随机伊辛模型的谱隙和收敛到平衡态的速度。 (5)Potts模型，其中外部场和边界条件彼此相反。 (6)利用渗流的思想来创建一个模型，该模型模仿了含有杂质的水的冻结的某些特征。 该项目通过一名博士的支持，帮助人力资源开发。这项工作是数学家和物理学家正在进行的努力的一部分，以了解自然世界中的各种系统，其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）穿过不规则材料的波，例如穿过地壳的地震波;（iii）半导体中的杂质;（iv）具有特殊性质的复合材料;以及（v）液体穿过多孔材料的渗透，例如水或油穿过地下岩石。长期以来，人们已经认识到，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深入了解现实世界的现象，这些抽象系统并不打算专门为磁铁、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的功能，如集群和临界现象。 亚历山大将要研究的系统--渗流、随机簇模型、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。

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 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 discrepancy between the actual and ideal shape of this droplet. (2) Percolation models also conditioned to exhibit the formation of a large droplet, with emphasis on lower bounds for the local fluctuations of the boundary. (3) Mixing properties and decay of correlations and connectivities for spin systems and percolation models, particularly in finite volumes. (4) The spectral gap and rate of convergence to equilibrium, for the stochastic Ising model at low temperature. (5) Potts models in which the external field(s) and the boundary condition are opposed to each other. (6) The use of percolation ideas to create a model which mimics certain features of the freezing of water which contains impurities. The project aids human resource development through the support of one Ph.D. student working on similar problems.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) composite materials with special properties; 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. The systems which Alexander will investigate--percolation, random cluster models, Ising and Potts models, and other spin systems--are examples of such abstract systems.