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.
During the course I will introduce the basic Bernoulli Percolation model, as well as the Ising and FK models and, at the first stage, study their basic properties in sub-critical and super-critical regime. This includes, among other properties, fundamental Menshikov’s theorem and Burton-Keane theorem on uniqueness of infinite cluster. At the second stage of the course I will focus on the critical and near-critical regime, where most of recent excitement lies. I will introduce exploration process and Stochastic Schramm-Loevner evolution (SLE), prove Smirnov’s theorem and will discuss several renormalization techniques including Barski-Newman method and fundamental Grimmett-Marstrand theorem).
Literature
Geoffrey Grimmett: Percolation, Second Edition; Grundlehren der mathematischen Wissenschaften, vol 321, Springer, 1999. Geoffrey Grimmett: The Random-Cluster Model; Grundlehren der mathematischen Wissenschaften Series, Springer, 2008. Vincent Beffara: Cardy’s formula on the triangular lattice, the easy way; Universality and Renormalization, vol. 50 of the Fields Institute Communications (2007), pp. 39-45. Vincent Beffara: Is critical 2D percolation universal? In and Out of Equilibrium 2 (2008), vol. 60 of Progress in Probability, Birkhäuser, pp. 31-58. Stanislav Smirnov: Conformal invariance in random cluster models. I; Holomorphic fermions in the Ising model, to appear in Ann. Math. Wendelin Werner: Lectures on two-dimensional critical percolation, arXiv:0710.0856.
Prerequisites
Knowledge of the basic first year (undergraduate level) probability.
Examination
There will be two tests during the course, based on which the evaluation will be made.