RUI: Statistical Mechanics Models on Non-Amenable Graphs

本文的目的是研究各种无限图上的几种随机空间模型。提出了自回避行走、渗透、伊辛和随机聚类模型以及伊辛模型的随机动力学等相关课题。对于自避行走，作者希望在不可服从图上回答模型的一些基本问题。这些问题包括：(1)n步自我回避步行的平均端到端距离是多少？(2)从给定点开始的n步自避免行走的数目的渐近行为是什么？渗透研究的主题是d维双曲图上模型的临界行为和临界指数，而d维双曲图是双曲d维空间的正则细分。不可服从图上的伊辛模型的主要课题涉及物理文献中具有重要基础意义的问题。它们包括：(1)在不同的温度区域有多少个极端吉布斯态？它们是什么？(2)有多少平动不变吉布斯态，它们是什么？提出了各种无限图上的Ising模型的随机动力学问题，其中自旋按照通常的Glauber动力学演化。所提出的课题来源于材料科学、统计物理、化学、生物学等科学领域，并具有广泛的应用。对自避行走的研究起源于化学物理学，作为长聚合物链的模型。它是表征聚合物链中单体之间的自排斥的最简单的数学模型，但却表现出临界现象。然而，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洛厄纳进化。渗透是研究通过无序系统的流动的概率模型，例如粒子流过防毒面具的过滤器，或者流体从随机多孔介质的间隙中渗出。Ising模型捕获了材料的基本磁性。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本质。提案人将继续努力让本科生承担这里提出的小问题。该计划如获资助，将支持学生在本科生研究/教育会议上展示他们的研究成果。

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.