Probability and Statistical Mechanics

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)伊辛模型通过固定少数自旋的过剩量来表现大液滴的形成，强调了这个过剩量的临界尺寸以及液滴实际形状与理想形状之间的差异。(2)渗流模型也以表现大液滴的形成为条件，强调边界局部波动的下界。(3)自旋系统和渗流模型的混合性质和关联和连通性的衰减，特别是在有限体积中。(4)低温下随机Ising模型的谱隙和收敛到平衡态的速率。(5)外场与边界条件相对立的波茨模型。(6)利用渗透的思想来创建一个模型，模拟含有杂质的水的某些冻结特征。该项目通过一名研究类似问题的博士生的支持，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；（四）特殊性能的复合材料；(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系统——渗透、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.