New Dimensions in Probability on Groups

点击翻译按钮获取中文摘要

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.