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 , which we view as a graph with edges between neighbouring vertices. All edges of are, independently of each other, chosen to be open with probability and closed with probability . 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 ?' A limit as of the question raised above is `What is the probability that there exists an open path from to infinity?' This probability is called the percolation probability and is denoted by . Clearly and , since there are no open edges at all when and all edges are open when . For some models there is a such that the global behaviour of the system is quite different for and for . 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.
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]
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]