Probability on Combinatorial Structures

0406017 Lyons PI正在调查各种组合结构的概率的几个领域的问题。这些问题中的许多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。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。其他行列式动力系统的相变和熵也在研究之中。耦合问题也是一些随机比较不等式研究的核心。基本问题涉及两种随机环境中随机游动行为的比较。 统计物理学领域在很大程度上涉及相变的数学模型（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，这取决于阻塞流动的颗粒的密度;随着密度的增加，存在相变，在某个点之后，概率为1，流体不再能流动任意远。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15年前，几位研究人员开始研究与我们最熟悉的、最接近我们物理世界的通常的欧几里得格完全不同的格。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8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。

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.