University of Geneve

We will discuss the percolation model on the hexagonal grid. In 2001 S. Smirnov proved conformal invariance of its scaling limit through the use of a tricky auxiliary combinatorial construction.

We present a more conceptual approach, implying that the construction in question can be thought of as geometrically natural one.

The main goal of the talk is to make it believable that not all nice and useful objects in the field have been already found.

No background is required.

 In this talk, we describe how to compute the critical point for various lattice models of planar statistical physics. We will first introduce the percolation, Ising, Potts and random-cluster models on the square lattice. Then, we will discuss how critical points of these different models are related. In a final part, we will compute the critical point of these models. This last part harnesses two main ingredients that we will describe in details: duality and sharp threshold theorems. No background is necessary.