Mondays 8:15-9:45 and Wednesdays 15:15-17:00

Meeting place: Room S1.

since the questions are easily stated and easily understood but the solutions (when they

exist) are typically nontrivial and very interesting. Many of the themes which occur in the

subject of "phase transitions and critical phenomena" in statistical mechanics or in

interacting particle systems occur in percolation theory. This makes percolation theory

a good area in which to become acquainted to such ideas.

First lecture (written by Oskar Sandberg)

Second lecture (written by Oskar Sandberg)

1.3-1.4,2.1-2.4 (without proof of BK),3.2,8.1-8.3. We went through the proof

that $p_c(2)=1/2$ (based on handout, not based on the book).

Some discussion of the RSW theorem. I will go through Lemma 11.75 on page

317 but will leave the harder Lemma 11.73 for you to read. (I am also trying to get

together a simpler proof of this.) The square root trick. (See inequality at top

of page 289). (this is used in the hard part of RSW). Lemma 11.12 on page 288

which gives an alternative proof that there is no percolation at 1/2 in

Z^2. This proof uses the square root trick and uniqueness. All of Section 6.1 (except all of 6.16).

This section used the subadditive theorem on page 399 (which we proved) and Theorem 2.38 from

reliability theory (which we didn't prove). We also proved all i.i.d. processes are ergodic,

meaning any translation invariant has probability 0 or 1. (It was a picture proof; no proof in book).

Section 6.2. Theorem 6.78 in Section 6.3. Theorem 5.4 in Section 5.2. Theorem 6.75. Theorems 8.18 and 8.21.

Theorem 8.61. Theorem 8.92, Theorem 8.97 and Theorem 8.99.

Assignment 1 Due September 30

Assignment 2 Due October 21

Assignment 3 Due November 14th

This course is intended for graduate students in mathematics and mathematical statistics.

Faculty are also of course very welcome.

Some probability theory (ask me if you are unsure).

Percolation, 2nd edition by G. Grimmett

There will be some homeworks, a final oral exam and perhaps (depending on the

number of students) student presentations.

Jeff Steif (steif@math.chalmers.se)

Below is some preliminary list of what will be covered. This way you can read

ahead. I strongly recommend the book above; it is an extremely good book. I

also (of course) recommend reading ahead as you will of course learn more this way.

contains 2 proofs of a very important result mentioned in Section 5.1 (Theorem 5.4).

However, I want to do the proof later on in the course and assume the result for now since

it is quite technical.

of the "subcritical" regime), I thought it would be nice to break to one of the exciting results in

percolation: uniqueness of the infinite cluster. This doesn't require any of the previous

material, is very beautiful and proved in 8.2. In Section 8.3, we then use the uniqueness

to prove that the percolation function is continuous; this is also independent of (most of) the .

previous material.

we do the famous exact calculation that the critical value is 1/2 in 2 dimensions.

We might discuss some newer things such as

(a) percolation on other graphs.

(b) percolation on trees.

Last modified: Wednesday April 13 10:15:34 MET DST 2005