Discrete Spatial and Physical Processes in Probability
(Percolation and Interacting Particle Systems)
Wednesdays, beginning 1 October 2008
IHP, Amphi Darboux
11.00–12.30 and 14.00–15.30
No course on Wednesday 29 October
The course will be pitched at a level suitable for the M2 at
Paris VI and Orsay.
The intended audience includes
visitors to the IHP, and local mathematicians and students.
The notes for the course are available under the title
(Chapter 2:) Uniform spanning trees and forests.
Wilson's algorithm for generating UST, introduction to SLE.
(Chapter 1:) Random walks and electrical networks. Polya's theorem.
(Chapter 3:) Percolation and self-avoiding walks, introduced.
(Chapter 4:) Correlation inequalities (FKG, BK).
Influence and sharp-threshold theorems.
Back to percolation (Chapter 5). Outline of
the sub-and super-critical phases. Uniqueness of
Exact critical point in two dimensions.
Cardy's formula with proof.
No course today, back next week.
Further discussion of the percolation phases. Critical percolation,
exponents and scaling relations. Conjectures.
Moving on to the contact model for the spread of infection.
Contact model on a tree.
Gibbs states. Ising/Potts and random-cluster models.
- 19 November:
Random-cluster model: theory and open problems. Connection to Potts
model. Random-cluster representation of the quantum Ising model.
Continuation class at 1600 in Amphi Darboux for students.
- 26 November:
Finish the qIm. Interacting particle systems (exclusion/voter models).
Random graphs. Lorentz gas.
will be at 0930 on 10 December, and will last
three hours. Room: 314 on third floor of the IHP.
The language of the exam will be French, and solutions may
be written in either language.
Some past Cambridge examination papers:
Cambridge Part 3 exam (2003)
Cambridge Part 3 exam (2006)
Cambridge Part 3 exam (2007)
This is just for information. The
Paris examination will
also be an m/n examination (read the rubric &mdash
the instructions on the exam paper!),
and may well contain an essay question.
I would expect the questions otherwise to be more `in the Paris style'.
Please email me (grg@statslab etc) in case of any problem.
GOOD LUCK!, and I hope you
enjoyed the course as much as I did.