New Dimensions in Probability on Groups

渗流理论是一个有60年历史的领域，有着许多分支。它最初是由对统计力学感兴趣的物理学家提出的，它与伊辛模型密切相关，伊辛模型从数学上描述了磁现象。与大多数物理模型一样，数学家们很早就对证明渗流理论中的严格结果感兴趣。即使是最简单的渗流实例，也就是二维晶格的特例，也涉及一些深层次的问题，如标度极限的共形不变性，斯米尔诺夫因此而在2010年被授予菲尔兹奖。在过去的几十年里，统计力学中通常研究的随机过程，如上述物理现象的模型，正在更抽象的设置中进行研究，例如，在展示类似于埃舍尔的一些图形的双曲几何的晶体中。这类过程研究最多的例子是随机游动和渗流。最基本的问题是，基本的“晶体”(即群的Cayley图)的代数或几何性质如何与其上随机过程的统计行为有关。在这个方向上的开创性结果是凯斯滕的定理，即当且仅当晶体(基团)是不可服从的时，随机游走返回概率按指数衰减。关于渗流的一个类似的著名结果指出，群G是不可服从的当且仅当存在一个G不变的渗流模型，该模型显示了巨团簇的唯一性和非唯一性之间的相变。许多相似性质的结果在概率论、几何学和群论的交叉点上形成了一个迅速发展的领域，可以被描述为‘群上的概率’。这一系列工作加深了我们对所讨论的随机过程的理解，甚至在统计力学中感兴趣的标准特殊情况下也是如此，它对进一步的研究方向如随机几何也起到了催化作用。当前项目设定的主要目标是将这一领域进一步发展，将其确立为几何群论中的工具，更广泛地说，是度量几何中的工具。如果所涉及的随机性质比群论性质更容易处理，那么上述定理作为工具将更有价值。但到目前为止，这种情况几乎没有发生过，我们的目标是改变这种情况。我们引入了新的不变量-‘维’的新概念，它们是用随机过程定义的，在群论运算方面表现良好，并解释了它们如何被用来证明群论的陈述。因此，本项目的愿景是将关于群的广泛而深入的工作转化为攻击群论问题的武器库。

Percolation theory is a 60-year old area with many ramifications. It was initially introduced by physicists interested in statistical mechanics, and it is closely related e.g. to the Ising model which mathematically describes the phenomenon of magnetism. Like with most physical models, mathematicians took an early interest in proving rigorous results in percolation theory. Even the simplest instance of percolation, namely the special case of the 2-dimensional lattice, entails deep questions such as conformal invariance of the scaling limits, for the proof of which Smirnov was awarded the Fields medal in 2010.In the last decades, stochastic processes typically studied in statistical mechanics as models of physical phenomena such as the above are being studied in more abstract setups, for example in `crystals' displaying hyperbolic geometry similar to some of Escher's figures. The most studied examples of such processes are random walks and percolation. The fundamental question is how the algebraic or geometric properties of the underlying `crystal' (i.e. Cayley graph of a group) relate to the statistical behaviour of the random process on it. The pioneering result in this direction is Kesten's theorem that the random walk return probability decays exponentially if and only if the crystal (group) is non-amenable. An analogous well-known result for percolation states that a group G is non-amenable if and only if there is a G-invariant percolation model displaying a phase transition between uniqueness and non-uniqueness of the giant cluster.Many more results of similar flavour form a rapidly developing area at the intersection of probability, geometry and group theory, that can be described as 'Probability on Groups'. This body of work has deepened our understanding of the stochastic processes in question even in their standard special cases of interest in statistical mechanics, and it has been catalytic in further directions of research such as Random Geometry. The prime objective set by the current project is to take this field a step further by establishing it as a tool in Geometric Group Theory, and more generally in Metric Geometry. Theorems like the above would be more valuable as tools if the stochastic property involved was easier to work with than the group theoretic one. But so far this is hardly ever the case, and our aim is to change this situation. We introduce new invariants ---new notions of `dimension'--- that are defined using stochastic processes and behave well with respect to group-theoretic operations, and explain how they can be used to prove group-theoretic statements.Thus the vision of this project is to turn the wide and deep body of work on Probability on Groups into an arsenal for attacking group-theoretic problems.