CO905 Stochastic models of complex systems

Online Course Materials

Lecturer: Stefan Grosskinsky

Lectures: Thu 12-13 and Fri 10-11 in D1.07

Classes: Fri 14-15 in D1.07

Tutorials: Fri 15-16 in D1.07 (by Paul Chleboun)

Vivas: Thu 17.3. and Fri 18.3. in B1.16 in Maths (schedule)

checklist for viva preparation: contents.pdf

Notes

• Final version of notes (March 15, typos in (1.32), Thm 1.14 and (2.14) corrected, thanks for pointing them out!): notes_co905_11.pdf
• Last year's course notes: notes_co905_10.pdf
(The syllabus has changed compared to last year and these notes only provide a very rough idea.)
See here for all of last year's online materials.
• For students with a good knowledge in probability the notes of the course MA4H3 have useful background material for later parts of the course.

Changes/Related Events

• Last lecture Thursday 17.3. at 12 (revision)
Vivas are on Thursday afternoon and Friday.
• Friday 25.02. there is no computer tutorial from 15-16, only the usual class from 14-15.
• MIR@W day on Monte Carlo Methods on March 7
• First lecture on Thu 13.01.2011

Problem Sheets

• sheet3: Voter model, scaling, Brownian motion, Contact process (due 07.03 at 11.30, 36/100 marks)
Remarks: (pdf has been updated, sorry for the large amount of typos...)
- Q3.4: (a) Example plot only to get an idea, please use increments of 0.01 for lambda as described in the question and plot more values.
- Q3.4: marks changed to 12 for (a) and 6 for (b)
- Q3.4: better use powers of 2 for system sizes, i.e. L=64,128,256,512 in (a) and L=128 in (b)
- Q3.3: typos: replace Z by \xi in (a), and B^1, B^2 by B and \tilde B in (b)
- Q3.1: in (c) there is a factor 2 missing on the RHS
- Q3.1: in (d), replace d/ds by d^2/dx^2 on the right-hand side and add factor of 2

• sheet2: Urn models, contact process, exclusion processes (due 15.02., 36/100 marks)
Remarks:
- 2.1(a): 'uniformly' means 'uniformly among the balls' not uniformly among the urns, so to make up the Markov chain, pick one ball, then what is the probability for it being in urn 1?

• sheet1: Generators/eigenvalues, branching processes, Toom's model (originally due 26.01., 28/100 marks)Remarks and errors:
- 1.1(c)*: Derive an 'implicit equation' rather than a 'recursion relation'
- 1.2(b): The equation for s* cannot be solved explicitly.
- 1.3(c): The walk is not irreducible (will be defined in lectures next week) and has many stationary distributions. Just explain the different possibilities for the long-time behaviour of the system.

Handouts

• handout6: Proof of Thm 3.5 (non-examinable)
• handout5: Connection between stochastic particle systems and PDEs (done for TASEP and Burgers equation)
• handout4: Poisson process, random sequential update, exponentials
• handout3: Characteristic function, Gaussians, LLN, CLT
• handout2: Some background on linear algebra
• handout1: Generating functions (with kind permission of C. Goldschmidt)

Suggested Books

• Gardiner: Handbook of Stochastic Methods (Springer).
• Grimmett, Stirzaker: Probability and Random Processes (Oxford).
• Grimmett: Probability on Graphs (CUP). (available online here)