To give an introduction to percolation theory, and study some of the newest developments.
Percolation theory deals with the connectivity properties of large, possibly infinite, networks (for instance a hexagonal lattice) from which a certain fraction q of the nodes or bonds is randomly removed. It is inspired by phenomena in physics and life sciences, but has become a mathematical topic of independent interest. It provides one of the mathematically most elegant examples of critical behaviour: there is a critical value of the parameter q at which the global properties of the system change drastically.
The first part of the course gives a general introduction and treats several classical results, in particular the uniqueness of the infinte cluster (in any dimension) and a proof that the critical probability for bond percolation on the square lattice is 1/2. Then we turn to more recent exciting developments, which started around 2000 and where work by Fields medalist Stanislav Smirnov on conformal invariance plays a key role. Finally we discuss current research and open problems, including the question of absence of percolation at the critical point. This is known for dimension 2 and for sufficiently high dimensions. For other dimensions (e.g. 3) this is one of the biggest current open problems.
Basic knowledge of probability and analysis. Some knowledge of conformal maps is useful in the second half of the course, but not necessary (what we use will be introduced and explained).
- Docent: Rob van den Berg