# Percolation Theory (5 points)

## NEW Meeting times: Mondays 8:15-9:45 and Wednesdays 15:15-17:00 Meeting place: Room S1.

Percolation theory is a beautiful subject within probability theory. It is very attractive
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.

• Lecture notes for course. (This will not be done for every lecture).

First lecture (written by Oskar Sandberg)
Second lecture (written by Oskar Sandberg)

• What have we covered so far?

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.

• Homeworks are following.

Assignment 1 Due September 30

Assignment 2 Due October 21

Assignment 3 Due November 14th

INTENDED AUDIENCE:
This course is intended for graduate students in mathematics and mathematical statistics.
Faculty are also of course very welcome.

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

Course literature:
Percolation, 2nd edition by G. Grimmett

Examination Form:
There will be some homeworks, a final oral exam and perhaps (depending on the
number of students) student presentations.

Kursexaminator:
Jeff Steif (steif@math.chalmers.se)

WHAT WILL BE COVERED:
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
• Chapter 1. Read through all.
• Chapter 2. Read through all (although we might not do all of 2.5 and 2.6).
• Chapter 3. Read Section 3.2.
• Chapter 4. Probably won't do any of this.
• Chapter 5. Read Section 5.1 and theorem 5.4 in Section 5.2. The rest of this chapter
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.
• Chapter 8. Read Sections 8.2 and 8.3. Before continuing with Chapter 6 (which is an analysis
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.
• Chapter 6. We will go through most of this: it covers pretty well the subcritical regime theory.
• Chapter 8. We will go through all of this: it covers pretty well the supercritical regime theory.
• We might do some parts of Chapter 7 which will introduce the important technique of renormalization.
• Chapter 11. This covers a lot of the theory for 2 dimensional percolation. In particular,
we do the famous exact calculation that the critical value is 1/2 in 2 dimensions.
• Chapter 5. The Aizenman-Barsky proof of the main result in Chapter 5 mentioned above.

We might discuss some newer things such as
(a) percolation on other graphs.
(b) percolation on trees.