MA3H2 Content

Content:

Let us briefly explain the mathematical setting using the example of Bernoulli percolation. Percolation is a simple probabilistic model which exhibits a phase transition. The simplest version of percolation takes place on $\mathbb{Z} ^2$ , which we view as a graph with edges between neighbouring vertices. All edges of $\mathbb{Z}^2$ are, independently of each other, chosen to be open with probability $p$ and closed with probability $1-p$ . A basic question in this model is `What is the probability that there exists an open path from the origin to the exterior of the square $S_n=[-n,n]^2$ ?' A limit as $n\to\infty$ of the question raised above is `What is the probability that there exists an open path from $0$ to infinity?' This probability is called the percolation probability and is denoted by $\theta(p)$ . Clearly $\theta(0)=0$ and $\theta(1)=1$ , since there are no open edges at all when $p=0$ and all edges are open when $p=1$ . For some models there is a $0<p_{c}<1$ such that the global behaviour of the system is quite different for $p<p_{c}$ and for $p>p_{c}$ . Such a sharp transition in global behaviour of a system at some parameter value is called a phase transition or a critical phenomenon, and the parameter value at which the transition takes place is called a critical value.

Books:

We will not follow a particular book. However, there are several sets of lecture notes used in the course, which can be downloaded from the Moodle page. The list below is a selection of books for a much deeper study of the subject.

H.O. Georgii: Stochastics: introduction to probability theory and statistics, de Gruyter (2008). [basic introduction to stochastics and Markov chains (discrete time)]

J. Norris: Markov chains, Cambridge University Press [standard reference treating the topic with mathematical rigor and clarity, and emphasizing numerous applications to a wide range of subjects]

G. Grimmett, D. Stirzaker: Probability and Random Processes, OUP Oxford (2001) [chapter 6 on Markov chains]

G. Grimmett: Probability on Graphs, Cambridge University Press (2010). [Available Online, contains a nice introduction to processes on graphs and percolation]

B. Bollabás, O. Riordan: Percolation, Cambridge University Press (2006). [a modern treatment of percolation. The introduction and the chapter on basic techniques are relevant for the lecture]

G. Grimmett: Percolation, 2nd ed., Springer (1999). [the standard reference on percolation. It contains much more than covered in the lecture. The first two chapters are relevant for the lecture]