RUI: Statistical Mechanics Models on Non-Amenable Graphs

RUI：不适合图的统计力学模型

• 批准号: 0505484
• 负责人: Chris Wu
• 金额: $ 6.39万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505484&HistoricalAwards=false
• 关键词: RUI; Statistical; Mechanics; Models; Non

The aim of this proposal is to study several random spatial models on various infinite graphs. Several related topics concerning self-avoiding walks, percolation, Ising and random cluster models, and stochastic dynamics of Ising models are proposed. For the self-avoiding walk, the proposer hopes to answer some fundamental questions of the model on non-amenable graphs. Such questions include: (1) What is the average end-to-end distance of an N-step self-avoiding walk? (2) What is the asymptotic behavior of the number of N-step self-avoiding walks starting from a given point? The topic in percolation concerns the critical behavior and critical exponents of the model on d-dimensional hyperbolic graphs, which are regular tessellations of the hyperbolic d-dimensional space. The main project of Ising models on non-amenable graphs concerns problems which are of fundamental importance in physics literature. They include: (1) How many extremal Gibbs states are there in different temperature region? What are they? (2) How many translation-invariant Gibbs states are there and what are they? Problems concerning stochastic dynamics of Ising models on various infinite graphs where spins evolve according to the usual Glauber dynamics are also proposed.The proposed topics originate from and have a wide range of application in materials science, statistical physics, chemistry, biology and other fields of sciences. The study of the self-avoiding walk arose in chemical physics as a model for long polymer chains. It is the simplest mathematical model which characterizes the self-repulsion between monomers in a polymer chain, yet exhibits critical phenomena. The interest in this model is, however, much broader since it is closely related to other stochastic models such as the Brownian motion and the newly discovered stochastic Loewner evolution. Percolation is a probabilistic model of studying flow through a disordered system, such as particles flowing through the filter of a gas mask, or fluid seeping through the interstices of a random porous medium. Ising models capture the basic magnetic properties of materials. An important goal of this project is to understand how the geometry of the space affects the nature of macroscopic behavior of these models. The proposer will continue to make an effort to involve undergraduate students to undertake small pieces of the problems proposed here. The proposal, if funded, will also support the students to present their findings in undergraduate research/education conferences.

该方案的目的是研究各种无限图上的几种随机空间模型。提出了与自回避行走、渗流、伊辛模型和随机簇模型以及伊辛模型的随机动力学有关的几个问题。对于自回避行走，作者希望回答该模型在非服从图上的一些基本问题。这样的问题包括：(1)N步自回避行走的平均端到端距离是多少？(2)从给定点开始的N步自回避行走次数的渐近行为是什么？渗流中的主题涉及模型在d维双曲图上的临界行为和临界指数，这是双曲d维空间的正则格子。关于不可服从图的Ising模型的主要项目涉及在物理文献中具有重要意义的问题。它们包括：(1)在不同的温度范围内有多少个极值吉布斯态？它们是什么？(2)有多少个平移不变的吉布斯态？它们是什么？还提出了各种无限图上Ising模型的随机动力学问题，其中自旋的演化符合通常的Glauber动力学。所提出的课题起源于材料科学、统计物理、化学、生物等科学领域，并具有广泛的应用。自回避行走的研究起源于化学物理，作为长聚合物链的模型。这是最简单的数学模型，它描述了高分子链中单体之间的自排斥，但表现出临界现象。然而，人们对这个模型的兴趣要广泛得多，因为它与其他随机模型密切相关，如布朗运动和新发现的随机Loewner演化。渗流是一种研究无序系统流动的概率模型，如通过防毒面具过滤器的颗粒流动，或通过随机多孔介质的缝隙渗流的液体。伊辛模型捕捉到了材料的基本磁性。这个项目的一个重要目标是了解空间的几何形状如何影响这些模型的宏观行为的性质。提倡者将继续努力让本科生承担这里提出的小部分问题。这项提案如果得到资助，还将支持学生在本科生研究/教育会议上展示他们的发现。