Phase transitions are natural phenomena in which a small change in an external parameter, like temperature or pressure, causes a dramatic change in the qualitative structure of the object (e.g., water boils at 100 degrees Celsius). To study this, many scientists (such as Nobel laureates Pauling and Flory) proposed the abstract framework of lattice models: the material is modeled as a collection of particles on a regular lattice, interacting (probabilistically) only with their nearest neighbors. In spite of the simplistic nature of this description, lattice models have proven to be a rich laboratory for the mathematical study of phase transitions. Since the revolutionary work of Schramm in 2000, the probabilistic approach to the study of these models has yielded a veritable explosion of new insights, with two Fields medals being awarded to Smirnov and Werner for their breakthroughs.

In this course, we aim to familiarize the audience with a modern approach to the probabilistic theory of lattice models, using Bernoulli percolation and the Ising model as our main examples. We then apply the theory to establish some very recent results on the study of random Lipschitz functions.

A particular focus will be given to two-dimensional models, where even the simplest models lead to a dazzling array of different fractal behaviors. This is a consequence of the conformal invariance of these models, which is predicted for all the models discussed, but rigorously proved in very few cases. One of our goals is to present Smirnov's proof of the conformal invariance of critical site percolation on the triangular lattice.

Prerequisites: the course Introduction to Probability.

Lectures will be in English.