Introduction to Percolation and Geometry of Groups
M. LI Jhih-Huang
Ecole Normale Superieure, France.
Percolation is a very simple model in statistical mechanics, but only a few results are known and proven nowadays. Given a graph, we open every edge with probability p and close it with probability 1-p, every edge being independent. Then, we would like to understand the "typical" structure of the resulting subgraph given by open edges. In fact, we know that there exist some critical values of probability which separate completely different structures.
In this lecture, I will introduce the model of percolation first. Some results and useful techniques (insertion tolerance, FKG inequality, etc.) will be presented. In particular, we will concentrate on percolation on Z^d and on trees, where the behaviors are quite different. At the end, some relations between geometry of groups and percolation will be mentioned.
Only basic notions in probability theory are required. The aim of this lecture is to give an introduction to this probabilistic model and to give some intuitions.