Introductory description

This module provides an introduction to continuous-time Markov processes and percolation theory, which have numerous applications: random growth models (sand-pile models), Markov decision processes, communication networks.
The module first introduces the theory of Markov processes with continuous time parameter running on graphs. An example of a graph is the two-dimensional integer lattice and an example of a Markov process is a random walk on this lattice. Very interesting problems of such processes involve spatial disorder and dependencies (e.g. burning forests). Therefore, after the main part, an elementary introduction to percolation theory will be given which can be used to study such questions.
Percolation is a simple probabilistic model for spatial disorder, and in physics, chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials. Recent applications include for example percolation of water through ice which is important for the melting of the ice caps.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Fundamental concepts of probability theory such as probability space and random variables. Random walks on graphs, and the related theory of electrical networks. Markov Chains, and their convergence to a stationary distribution. Galton-Watson trees. Percolation theory: the phase transition, and determining the threshold in particular cases.

Learning outcomes

By the end of the module, students should be able to:

• The module will enhance the students' knowledge of a foundational core of pure mathematics and an understanding of a range of techniques.
• Research skills will be developed as well in the final two weeks.

Indicative reading list

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]

Subject specific skills

The module will enhance the students' ability to think clearly, learn new ideas quickly, and construct and follow logical arguments.

Transferable skills

Following complex reasoning, and manipulating precise and intricate concepts.