Prerequisites
Bachelor mathematics or physics.
Aim of the course
To give an introduction to percolation theory, and study some of the recent 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 was originally motivated by phenomena in physics and biology, 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 important classical results, in particular the uniqueness of the infinite cluster (in any dimension) and a proof that the critical probability for bond percolation on the square lattice is equal to 1/2. Then we turn to more recent 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 open problems
Lecturer
Rob van den Berg (VU)
- Docent: Rob van den Berg