Percolative systems are random spatial processes that have a huge number of locally interacting components, and where interesting “global” phenomena can be expressed in terms of paths of local events that percolate’ through space (or space-time). The interest in these systems has enormously increased during the last decade. Standard examples are models of porous materials, spatial epidemics, growth processes and ad-hoc wireless networks. Examples of phenomena where percolation plays a more subtle role are magnetization or localization of waves. The models usually have several parameters and an important problem is whether there is a phase transition (a dramatic change of the global behaviour at somecritical’ choice of the parameters), and how the system behaves at and near the critical point.
The main emphasize is on the basic Bernoulli percolation model, though we also touch upon Ising- and FK-models. At the first stage, we study their basic properties in sub-critical and super-critical regime. This includes, among other properties, the sharpness of phase transition and the Burton-Keane theorem on uniqueness of the infinite cluster. At the second stage of the course I focus on the critical regime, where most of recent excitement lies. For percolation on two-dimensional lattices, I introduce the exploration process and discuss its convergence to Schramm-Loewner evolution (SLE). Finally, I explain how lace expansion can be used to analyze critical behaviour for percolation on high-dimensional lattices.
Literature
Geoffrey Grimmett: Percolation, Second Edition, Springer, 1999. Wendelin Werner: Lectures on two-dimensional critical percolation, arXiv:0710.0856. Geoffrey Grimmett, Probability on Graphs, Cambridge University Press, 2010.
Prerequisites
Knowledge of first year (undergraduate level) probability
Examination
Final exam, written
Links
Course homepage