DNA-L, Spring 2010

In this talk I will introduce the audience to the vast topic of percolation. There are two major types: site percolation and edge percolation. The most interesting case is the one of critical percolation (to be introduced). In this case, motivated by physical arguments, Cardy conjectured that as the mesh size vanishes, the percolation probability is given by a certain conformal invariant formula, which assumes a particular simple shape in Carlesson's formulation. We will show Smirnov's beautiful proof of this conjecture, which works however only in the particular case of triangular ("honeycomb") lattice.

Background reading: articles by Smirnov and Werner.