Statistical Physics on Groups and Determinantal Probabilities

PI正在调查几个离散概率领域的问题，这些领域往往具有令人惊讶的相互联系。这些问题大多是在群不变的背景下设置的，目标是了解群的几何或代数性质如何反映在过程的概率性质中。例如，在随机聚类模型中，每个q值都有4个p的自然临界值。PI继续他之前在群的平面Cayley图上研究这些值之间的关系。正在研究的另外两个模型涉及图中的随机跨越森林。一个是由有限图的最小生成树的极限得到的，另一个是由一致生成树得到的。前者与渗透有关，这是随机聚类模型的一种特例。后者与随机游走和势能理论相联系，因此更容易被理解。PI致力于使最小生成森林的知识状态更接近于均匀生成森林的知识状态。PI正在调查的统一跨越森林也有许多悬而未决的问题。当一个人将均匀跨越森林视为决定性的概率度量时，就会出现大量的新问题。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，并建立通过类似于渗透而产生的猜想。其他决定性动力系统的相变和熵也在研究之中。统计物理学领域在很大程度上与相变（例如，水到冰）的数学模型有关。典型的空间模型是一个固定的晶格，例如，二维的方形晶格或三维的立方晶格。这个晶格是无限的，并且具有所谓群的数学性质。最简单的模型，被称为渗透，起源于对地下流体流动和气体通过防毒面具流动的研究。有人问流体可以流动多远，特别是，它是否可以任意流动。当然，这取决于粒子的密度；随着密度的增加，存在一个相变，即在某一点之后，流体不能再任意流动，概率为1。人们想知道那个点在哪里以及当这个临界点接近时概率是如何变化的。大约十年前，几位研究人员开始研究与我们最熟悉、最接近我们的物理世界的欧几里得晶格截然不同的晶格。这些新的格，被称为不可服从格，通常也是基于群的。这样的研究开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，即不可服从群体的统计物理学，引起了人们的极大兴趣。这一领域的研究内容相当丰富，包含了大量重要的基本问题，但其答案仍然未知。欧几里得格已经应用了一些新思想，这些新思想是为了响应为不可服从的群体开发新工具的需要而产生的。

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.