Statistical Physics on Groups and Determinantal Probabilities

群和行列概率的统计物理

• 批准号: 0103897
• 负责人: Russell Lyons
• 金额: $ 10万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103897&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正在调查几个离散概率领域的问题，这些领域往往有令人惊讶的相互联系。 这些问题中的大多数都是在组不变的上下文中设置的，目标是了解组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，有4个自然的p临界值。 PI正在继续他以前的调查，这些价值观之间的关系，平面凯莱图的群体。 其他两个模型正在调查中的关系图随机生成森林。 其中一个是从有限图的最小生成树的极限中得到的，而另一个是从一致生成树中得到的。 前者与渗流有关，这是随机簇模型的一个特例。 后者与随机游动和势理论有关，我们对它的理解要好得多。 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。 PI正在研究的均匀生成森林也有许多悬而未决的问题。 当人们把均匀生成森林看作是行列式概率测度时，就会出现大量的新问题。 例如，PI正致力于建立随机森林的高维类似物的基本拓扑性质，并建立由渗流类比产生的拓扑结构。其他行列式动力学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与相变的数学模型有关（例如，水冰）。 通常，空间模型是固定的晶格，例如，二维的正方形晶格或三维的立方晶格。 这个格是无限的，并且具有所谓群的数学性质。 最简单的模型，称为渗流，起源于对地下流体流动和气体通过防毒面具流动的研究。 人们问流体能流多远，特别是它是否能流任意远。 当然，这取决于粒子的密度;随着密度的增加，存在相变，在某个点之后，概率为1，流体不再能流任意远。 人们想知道这一点在哪里，以及当接近这一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的、最接近我们的物理世界的通常的欧几里德格完全不同的格。 这些新的格，称为noncompliant，通常也基于群。这样的调查始于推动基础研究的通常的科学和数学好奇心。 在过去的5年里，这一研究领域，统计物理学上的不服从群体，已经看到了爆炸的兴趣。 这一领域的研究结果是相当丰富的，并包含了大量的重要的基本问题，其答案仍然未知。 由于需要为不服从的群开发新的工具，一些新的思想已经被应用于欧几里得格。