Probability and Discrete Structures

PI正在研究不同组合结构上几个概率领域的问题。这些问题中的许多都设置在群不变的背景下，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，并建立类似于渗流的猜想。其他行列式动力系统的相变和熵也在研究中。耦合问题也是正在研究的一些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行走行为的比较。此外，PI正在通过调查有根网络上的所有单模概率度量是否都是软的来调查是否所有的群都是SOFIC的。统计物理领域在很大程度上涉及相变(例如，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，这取决于阻碍流动的颗粒的密度；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20年前，几位研究人员开始研究与通常的欧几里得格子完全不同的格子，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10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 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 intwo random environments. In addition, the PI is investigating whether all groups are sofic byinvestigating whether all unimodular probability measures on rooted networks are sofic.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 densityincreases, 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 20 years ago, several researchers began investigating lattices that are quitedifferent 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 10 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 arisenin response to the need to develop new tools for nonamenable groups.