Percolation and Geometry

Lecture:

Monday, 12-14, M1
Thursday, 12-14, M1

The aim of this course is to explore percolation theory and its role in understanding the geometry of Cayley graphs, as well as more general classes of graphs. Percolation is a fundamental stochastic process with deep connections to statistical mechanics; however, it has also recently emerged as a powerful tool for analyzing subtle geometric properties of graphs and groups. In particular, it provides insight into notions such as amenability, Kazhdan’s property (T), the Haagerup property, and hyperbolicity.

The course will begin with the basic concepts of percolation theory. A central example will be percolation on trees, which serves as a guiding case for developing intuition. From there, we will extend the discussion to more general transitive graphs and introduce key tools such as the mass transport principle. These techniques will be applied to study amenability and non-amenability, as well as to characterize important group properties, including property (T) and the Haagerup property. We will also examine the percolation-theoretic perspective on the notion of "cost" and "fixed price" for groups. 

This lecture in the course overview.

The tutorials in the course overview.