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。这些问题中的大多数都设置在群不变的上下文中，目的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，在随机簇模型中，q的每个值都有4个p的自然临界值。Pi继续他以前在群的平面Cayley图上对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生成森林。其中一个是从有限图的极小生成树的极限得到的，另一个是从一致生成树得到的。前者与渗流有关，这是随机团簇模型的一个特例。后者与随机游走和位势理论有关，理解起来要好得多。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状态。还有许多悬而未决的问题与PI正在调查的统一跨越森林有关。当人们将均匀分布的森林视为决定性的概率度量时，就会出现大量新的问题。例如，PI致力于建立随机森林的高维类似物的基本拓扑性质，并建立通过类比渗流而产生的猜想。其他行列式动力系统的相变和熵也在研究中。统计物理领域在很大程度上涉及相变(如水到冰)的数学模型。通常，空间模型是固定的点阵，例如，二维的正方形点阵或三维的立方点阵。这个格子是无限的，并且具有所谓群的数学性质。最简单的渗流模型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。有人问，流体能流到多远，特别是它是否能任意流得远。当然，这取决于粒子的密度；随着密度的增加，会有一个相变，在某一点之后，流体不能再以概率1的概率任意流动。人们想知道那个点在哪里，当接近这个临界点时，概率是如何变化的。大约十年前，几位研究人员开始研究与通常的欧几里得格子大不相同的格子，这些格子最熟悉，也最接近我们的物理世界。这些被称为不可服从的新格子通常也是基于群的。这类研究始于通常的科学和数学好奇心，这种好奇心推动了基础研究。在过去的5年里，这一领域的研究--关于不可服从群体的统计物理学--引起了人们的极大兴趣。事实证明，这一领域的研究相当丰富，包含了大量重要的基本问题，这些问题的答案仍然未知。已经有一些新思想应用于欧几里得格子，这些新思想是为了响应为不可服从群开发新工具的需要而出现的。

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.