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Markov Processes and Percolation theory 2010

Lecturer: Stefan Adams TA for the support class: William Nollett

Schedule: Monday 10-11 am room B3.03; Tuesday 3-4 pm room MS.03; Thursday 5-6 pm room MS.04

Support class starts week 2; Thursday 11-12, B1.01; Revision class Friday 7th May, 10-12 am room B3.02

Assessment: exam (85 %), example sheets (15 %). There are 6 example sheets (sheet 1,2,3,5 for credit), and sheet 6 is a Mock exam which is very close to the exam in term 3.


Programme

week 1: Discrete time random walks and basics in probability

week 2: Markov property and transition function; Poisson process; random walks

week 3: Q-Matrix and its exponentials

week 4: Forward and backward equations

week 5: Birth processes; hitting times; recurrence and transience

week 6: Invariant distribution

week 7: Ergodic theorem, convergence to equilibrium

week 8: Introduction to percolation

week 9: Peierl's argument; critical value; FKG inequality; Harris Theorem

week 10: Ruso-Seymour-Welsh Theorem/method (crossing of rectangles); Kesten's Theorem; p_c=1/2 for bond percolation on Z2




Suggested books

D.W. Stroock: An Introduction to Markov Processes, Springer (2005).

J. Norris: Markov chains, Cambridge University Press (1997).

G.F. Lawler, L.N. Coyle: Lectures on Contemporary Probability, American MAthematical Society (2000).