Percolation : from Introduction to Frontiers of Current Research, Spring 2018


General

This page gives some General Information on this Mastermath/WONDER course, and each week I will indicate here which material has been (planned to be) treated.



Literature used

We use parts of the book `Probability on Graphs' by Geoffrey Grimmett (Cambridge University Press), as well as lecture notes and some literature available from the internet. The course is meant for master students, but PhD students and postdocs are also welcome to attend. The lectures take place at the VU University on Thursdays, 10:00 - 12:45. The lecture room changes from week to week (see under "Material treated" below).


Material treated

Here I will mark each week which material I have treated (or plan to treat).

week 6 (Thursday February 8; room HG-02A00 at the VU): I plan to give an introduction and informal overview of the course, and discuss some preliminary results needed in the proof that the critical probability for bond percolation on the square lattice is equal to 1/2.

week 7 (Thursday February 15; room WN-C638 at the VU). I plan to continue with Section 2 in my LECTURE NOTES and, along the way, also discuss Section 1. Then I will start with Section 3. Here are the ILLUSTRATIONS referred to in the notes.

week 8 (Thursday February 22; room WN-Q112 at the VU) During the first two lecture hours, and the beginning of the third hour, I plan to discuss Section 3 of my lecture notes in more detail, and start with Section 4. In the third lecture hour I will also present a problem/exercise. (This exercise is purely to learn from, it has no effect on the grade). The third hour is also meant for asking questions about this exercise or the material treated so far in the course.

Week 9 (Thursday March 1; room WN-C659 at the VU) During the first two hours I plan to treat Sections 4, 5 and 6 of my lecture notes. In the third hour we will discuss the exercises I handed out in week 8.

Week 10 (Thursday March 8; room WN-C659 at the VU) Last week (March 1) I treated Section 4 and part of Section 5 of my lecture notes (up to and including the proof of Lemma 5.2). On March 8 I will complete Section 5 and Section 6, and start with a new sub-topic. In the third hour I will continue the discussion of the exercises handed out in week 8.

Week 11 (Thursday March 15; room WN-M129 at the VU) Last week (March 8) I discussed the remaining part of Section 5 in my Lecture Notes (starting with and including Proposition 5.3) and the short Section 6 (about uniqueness of the infinite open cluster). Then I presented the Burton-Keane proof for uniqueness in arbitrary dimension d \geq 2 (this corresponds with Section 5.3 in Grimmett's book Probability on Graphs (2010)). Part of the third hour was used to discuss problems. On March 15 we start with the famous proof (by Smirnov) of conformal invariance.

Week 12 (Thursday March 22; room WN-P631 at the VU) We continue with the proof of conformal invariance.

Week 13 (Thursday March 29; room WN-P631 at the VU) Plan is to complete the above mentioned proof and then start with a new subject.

Week 14 (Thursday April 5; room WN-P631 at the VU) Last week I completed Smirnov's proof of conformal invariance/Cardy's formula (Section 5.7 in Grimmett's book), and showed an application to connection probabilities in the half-plane. This week I will discuss some more applications, and then turn to near-critical percolation.

Week 15 (Thursday April 12; room WN-P631 at the VU) I will first return to some of the consequences (for the so-called one-arm half-plane exponent) of Smirnov's result (Cardy's formula), and then continue the discussion involving `characteristic length'.

Week 16 (Thursday April 19; I discussed a classical result by Harry Kesten on near-critical percolation.

Week 17 (Thursday April 26; room WN-M655 at the VU) We start the discussion of a new topic, in particular relevant for percolation in dimension 3 and larger.

Week 18 (Thursday May 3; room WN-P631 at the VU) We continue the discussion started last week.

Week 20 (Thursday May 17; room WN-P656 at the VU) We continue the discussion of the previous meeting

Week 21 (Thursday May 24; room WN-P631 at the VU) Last lecture of this course.


Last updates: May 23 2018