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MA5Q3 Topics in Complexity Science

Lectures: 2 hours in weeks 1-10 of term 2, time and place tbc

First lecture: tbc

Classes: 1 hour support class in weeks 2-10

Regularly updated course notes: will appear here

For inspiration for part of the covered material, you can check out the notes of the discontinued module CO905 Stochastic models of complex systems.

Assessment. The module is assessed 100% by essay which is due beginning of term 3, more information on topics and specifications can be found here.

Content. Stochastic particle systems are probabilistic models of complex phenomena that involve a large number of interacting components, such as the spread of epidemics or traffic flow. The module will focus on a mathematical description of simple minimal models and their large scale dynamic properties and fluctuations. In addition to basic mathematical theory we will also cover computational aspects and simulation techniques in the class.

Syllabus

  • Introduction and basic review of continuous-time Markov processes and examples (including birth-death chains, branching processes, Kingman's coalescent)
  • Generators, semigroups and description of fluctuations via martingales
  • Introduction to stochastic particle systems and agent-based models
  • General theory of large-scale emergent behaviour, criticality and phase transitions
  • Spin systems, Ising model, Markov chain Monte-Carlo (MCMC)
  • Disordered systems and applications to neuroscience, Hopfield model
  • Models of mass and energy transport and their scaling limits (including exclusion and zero-range processes)
  • Epidemic models and their scaling limits (including the contact process)
  • Sociological models of opinion formation (including voter model, Axelrod model)
  • Efficient simulation methods based on graphical constructions

Prerequisites. A background in continuous-time Markov chains is necessary, further background in measure theory, functional analysis and basic PDE theory can be useful but is not necessary.

Literature. Typed course notes will be made available.

  • Chapters 3 and 4 of: T.M. Liggett: Continuous Time Markov Processes. AMS Graduate Studes in Mathematics 113, 2010
  • T.M. Liggett: Interacting Particle Systems - An Introduction, ICTP Lecture Notes 17 (2004), http://publications.ictp.it/lns/vol17/vol17toc.html
  • C. Kipnis, C. Landim: Scaling Limits of Interacting Particle Systems. Springer, 1999
  • L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim: Macroscopic Fluctuation Theory. Reviews of Modern Physics, Volume 87, 593-636, 2015 (arXiv:1404.6466)