Probability in 2D
Fall 2016, Monday 16-19
Holzblat 007
Course outline
We will study some of the very recent progress in probability on
two dimensional lattices:
- A short new proof of Kesten's 1980 result: the critical
percolation probability on the square lattice is 1/2.
- The self-dual point of the random cluster model on the square
lattice is the critical point.
- The number of self-avoiding walks of length n on the hexagonal
lattice is (2+sqrt{2})^{n + o(1)}.
- The square crossing probability in Voronoi percolation is 1/2.
We will develop all the necessary tools such as various correlation
inequalities, sharp threshold theorem and noise-sensitivity of
boolean functions (for positively associated measures),
Russo-Seymour-Welsh techniques. We will assume first year
undergraduate knowledge in probability and analysis.
The course is partially based on the following papers:
The following books could also partially help:
- Percolation, by Geoffrey Grimmett.
- Percolation, by Bela Bollobas and Oliver Riordan.
- The random cluster model, by Geoffrey Grimmett.
Homework