New Dimensions in Probability on Groups

逾渗理论是一个有着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引入的，它与数学上描述磁性现象的伊辛模型密切相关。与大多数物理模型一样，数学家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，即二维晶格的特殊情况，也涉及诸如标度极限的共形不变性等深层次问题，为此Smirnov在2010年获得了菲尔兹奖。在过去的几十年中，通常在统计力学中研究的随机过程作为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“晶体”中，显示出类似于布里尔的一些图形的双曲线几何形状。这类过程研究最多的例子是随机游动和渗流。基本的问题是基本的“晶体”（即一个群的凯莱图）的代数或几何性质如何与其上随机过程的统计行为相关联。在这个方向上的开创性结果是Kesten定理，即随机游走返回概率指数衰减当且仅当晶体（群）是不顺从的。一个类似的著名的结果渗透状态，一个群G是non-affected当且仅当有一个G-不变的渗透模型显示之间的相变唯一性和非唯一性的巨型集群。更多的类似的味道形成一个快速发展的领域，在概率，几何和群论的交叉点，可以被描述为“概率群”。这一机构的工作加深了我们的理解的随机过程的问题，即使在他们的标准特殊情况下的兴趣在统计力学，它一直催化进一步的研究方向，如随机几何。当前项目的主要目标是通过将该领域建立为几何群论中的工具，以及更普遍的度量几何中的工具，将该领域进一步发展。如果涉及的随机性质比群论更容易处理，那么像上面这样的定理作为工具将更有价值。但到目前为止，这种情况几乎从未发生过，我们的目标是改变这种情况。我们介绍新的不变量-新的概念'维'-这是定义使用随机过程和表现良好的相对于群论的操作，并解释它们如何可以被用来证明群论statements.Thus的愿景这个项目是把广泛和深入的工作对群体的概率成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.