Holzblat 007

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.

The course is partially based on the following papers:

- The self-avoiding walk on the hexagonal lattice, by Smirnov and Duminil-Copin.
- The self-dual point of the two-dimensional random-cluster model is critical for q>=1, by Beffara and Duminil-Copin.
- A new proof of the sharpness of the phase transition for Bernoulli percolation on Z^d, by Duminil-Copin and Tassion.
- Quenched Voronoi percolation, by Ahlberg, Griffiths, Morris, and Tassion.

- Percolation, by Geoffrey Grimmett.
- Percolation, by Bela Bollobas and Oliver Riordan.
- The random cluster model, by Geoffrey Grimmett.

- Homework 1, due November 28th, 2016.
- Homework 2, due December 26th, 2016.