Critical lattice models and conformal invariance

Physicists were able to make a number of striking predictions about certain critical lattice models in the plane: percolation, self-avoiding walk, Ising model, …For example, it is expected that the number of distinct self-avoiding walks of length $N$ on any regular planar lattice grows like $M^N N^{11/32}$, where $M$ depends on the lattice, whereas 11/32 is universal! Similar rational numbers appear for other models: dimension of a critical percolation cluster is 91/48, while dimension of the frontier of a Brownian curve is 4/3. Recently there was substantial progress in mathematical understanding of these models, and their conjectured conformal invariance in the scaling limit, which plays the main role in the determination of dimensions.
We will introduce the mentioned models and discuss recent work of Lawler, Schramm, Werner, and the speaker, which led to the proof of some of the physics predictions.