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MA4K4 Content

Content:
1] Interacting Particle Systems
- Construction and definitions (graphical construction, semigroups and generators).
- Revision of basic concepts like stationary distributions and reversibility.
- Classical examples (to be used throughout the course).

2] Relaxation and Mixing Times
- Introduction and definitions of mixing times.
- Basic bounds and techniques.
- Spectral methods and relaxation times.
- Basics of Potential Theoretic approach, resistor networks for reversible Markov Processes.

3] Large Deviations
- Introduction with examples.

4] Metastability
- Application of Large Deviations.
- Application of Potential Theory.

Aims:
The principle aim is firstly to introduce basic stochastic models of collective phenomena arising from the interactions of a large number of identical components, called interacting particle systems. The module will then introduce several key topics which are currently at the forefront of mathematical research in interacting particle systems. In particular we fill focus on the study of large-scale dynamics.

Objectives:
By the end of the module the student should be able to:

  • Have a good working knowledge of key prototypical models of interacting particle systems such as the Ising model, the exclusion process and the zero-range process.
  • Understand the main concepts used in current research into the large scale dynamics of interacting particle systems.
  • Work in an independent and practical manner on topics related to interacting particle systems. Students should gain an advanced-level understanding of continuous time Markov processes on finite state spaces.
  • Build and run stochastic simulations using their preferred method (simple examples of C-code will be given, requiring straightforward adaptation, for those who do not have a strong background in this area). This module should also help students building team working skills.

Books:
- Levin, Peres, Wilmer: Markov Chains and Mixing Times, AMS (2009) [Available Online]
- T.M. Liggett: Interacting Particle Systems - An Introduction, ICTP Lecture Notes 17 (2004) [Available Online]
- F. den Hollander: Large Deviations, AMS (2000)
- A. Bovier: Metastability, Lecture notes - Prague (2006) [Available Online]
- Montenegro, Tetali: Mathematical aspects of mixing times in Markov chains (2006) [Available Online]