Statistical Physics on Groups and Determinantal Probabilities

群和行列概率的统计物理

• 批准号: 0231224
• 负责人: Russell Lyons
• 金额: $ 6.18万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0231224&HistoricalAwards=false
• 关键词: Statistical; Physics; Groups; Determinantal; Probabilities

The PI is investigating questions in several areas of discrete probability that often have surprising interconnections. Most of these questions are set in a group-invariant context and the goal is to understand how geometric or algebraic properties of the group are reflected in probabilistic properties of the processes. For example, in the random cluster model, there are 4 natural critical values of p for each value of q. The PI is continuing his previous investigations of the relations among these values on planar Cayley graphs of groups. Two other models under investigation concern random spanning forests in graphs. One of these is obtained from limits of minimal spanning trees in finite graphs, while the other is from uniform spanning trees. The former is connected to percolation, a special case of the random cluster model. The latter, connected to random walks and potential theory, is much better understood. The PI is working to bring the state of knowledge of the minimal spanning forest closer to that for the uniform spanning forest. There are also many open questions related to the uniform spanning forest that the PI is investigating. When one views uniform spanning forests as determinantal probability measures, there are a large number of new questions that open up. For example, the PI is working to establish basic topological properties of higher-dimensional analogues of random forests and to establish conjectures that arise by analogy to percolation. Phase transitions and entropy of other determinantal dynamical systems are also under investigation.The field of statistical physics is concerned to a great extent with mathematical models of phase transitions (e.g., water to ice). Typically the model of space is a fixed lattice, for example, the square lattice in two dimensions or the cubic lattice in three dimensions. This lattice is infinite and possesses the mathematical properties of what is called a group. The simplest model, known as percolation, originated in the study of fluid flow in the ground and gas flow through a gas mask. One asks how far fluid can flow, in particular, whether it can flow arbitrarily far. This, of course, depends on the density of particles; there is a phase transition as the density increases, whereby after a certain point, with probability 1, fluid can no longer flow arbitrarily far. One would like to know where that point is and how the probability changes as this critical point is approached. About a decade ago, several researchers began investigating lattices that are quite different from the usual Euclidean ones that are most familiar and that most closely correspond to our physical world. These new lattices, called nonamenable, are also usually based on groups. Such investigations began out of the usual scientific and mathematical curiosity that drives fundamental research. Within the last 5 years, this area of research, statistical physics on nonamenable groups, has seen an explosion of interest. This area of research turns out to be quite rich and to contain a large number of important fundamental questions whose answers remain unknown. Already, there have been applications to Euclidean lattices of some of the new ideas that have arisen in response to the need to develop new tools for nonamenable groups.

PI 正在研究几个离散概率领域的问题，这些领域通常具有令人惊讶的相互联系。 大多数这些问题都是在群不变的背景下设置的，目标是理解群的几何或代数属性如何反映在过程的概率属性中。 例如，在随机聚类模型中，对于每个 q 值，p 有 4 个自然临界值。 PI 正在继续他之前对平面凯莱群图上这些值之间关系的研究。 正在研究的另外两个模型涉及图中的随机跨越森林。 其中之一是从有限图中最小生成树的极限获得的，而另一个是从均匀生成树获得的。 前者与渗滤有关，渗滤是随机聚类模型的一种特殊情况。 后者与随机游走和势论相关，可以更好地理解。 PI 正在努力使最小生成森林的知识状态更接近均匀生成森林的知识状态。 还有许多与 PI 正在调查的统一跨越森林相关的悬而未决的问题。 当人们将均匀分布的森林视为决定性概率度量时，就会出现大量新问题。 例如，PI 正在努力建立随机森林的高维类似物的基本拓扑特性，并建立通过类比渗透产生的猜想。其他行列式动力系统的相变和熵也在研究中。统计物理学领域在很大程度上关注相变的数学模型（例如，水到冰）。 通常，空间模型是固定的格子，例如二维的方格子或三维的立方格子。 这个晶格是无限的，并且拥有所谓群的数学特性。 最简单的模型称为渗滤，起源于对地下流体流动和通过防毒面具的气体流动的研究。 人们问流体能流多远，特别是是否可以流任意多远。 当然，这取决于粒子的密度；随着密度的增加，会发生相变，在某一点之后，流体不能再以概率 1 任意流动。 人们想知道该点在哪里以及当接近该临界点时概率如何变化。 大约十年前，一些研究人员开始研究与我们最熟悉且最接近我们的物理世界的常见欧几里得晶格完全不同的晶格。 这些新的格子，称为不可命名的，通常也是基于组的。此类研究始于推动基础研究的通常的科学和数学好奇心。 在过去的五年里，无名群的统计物理学这一研究领域引起了人们的广泛兴趣。 事实证明，这一研究领域非常丰富，包含大量重要的基本问题，但其答案仍然未知。 一些新想法已经应用于欧几里得格，这些新想法是为了满足为无名群体开发新工具的需要而出现的。