Probability on Combinatorial Structures

[406017 . lyons] PI正在研究不同组合结构的几个概率领域的问题。这些问题中的许多都是在群不变的上下文中设置的，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。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。耦合问题也是一些正在研究的随机比较不等式的核心。基本问题涉及两个随机环境中随机漫步行为的比较。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，这取决于阻挡气流的颗粒的密度；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15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8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。

0406017Lyons The PI is investigating questions in several areas of probability on various combinatorial structures. Many 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. 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. Coupling questions are also at the heart of some stochastic comparison inequalities being studied. Basic questions concern comparison of the behavior of random walks in two random environments. 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 that block the flow; 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 15 years 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 8 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.