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
- 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.
- 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
- 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)
- 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.
- 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)
Matlab and C stuff
- Here is a new version of the contact process code using the Mersenne Twister: contact_mt-2.zip
You should have all files in the same directory.
- Simple C programs for the contact process: contact.c (for Q3.4(a)) and contact2.c (for Q3.4(b))
(as before, you might have to adapt the random number generator)
- Very simple C program traffic.c and Matlab file traffic.m for the TASEP (Q2.3 and Q2.4)
new version with output function for profiles traffic_new.c
- Matlab file for question 1.3 on Toom's model (by P. Chleboun): tooms.m
- Wikibooks on Matlab and C_Programming
- If you ever need a really good random number generator (not necessary for the module):
- Gardiner: Handbook of Stochastic Methods (Springer).
- Grimmett, Stirzaker: Probability and Random Processes (Oxford).
- Grimmett: Probability on Graphs (CUP). (available online here)
- tutorials on the Ising model, including Monte Carlo simulation methods: http://www.nd.edu/~mcbg/tutorials/2006/tutorial/ising.html, http://pages.physics.cornell.edu/sethna/teaching/Simulations/LMC.html
- if you are interested in Hidden Markov models and epsilon machines look here (unfortunately I could not find a good review which is available online).
- R.A. Blythe: Ordering in voter models on networks: exact reduction to a single-coordinate diffusion, J. Phys. A: Math. Theor. 43 385003 (2010)
- R.A. Blythe, A.J. McKane: Stochastic Models of Evolution in Genetics, Ecology and Linguistics, J. Stat. Mech.: Theor. Exp. (2007) P07018
R. Lambiotte, S. Redner: Dynamics of Non-Conservative Voters, http://arxiv.org/abs/0712.0364
T. Antal, P. L. Krapvisky, S. Redner: Dynamics of Microtubule Instabilities, http://arxiv.org/abs/q-bio/0703032
Frank Kelly: The Mathematics of Traffic in Networks