250093 VO Lattice Models (2022S)

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. To study this, many scientists (such as Nobel laureates Pauling and Flory) proposed the abstract framework of lattice models: the material was modeled as a collection of particles on a regular lattice, interacting only with their nearest neighbors. In spite of the simplistic nature of this assumption, 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 some classical results from the probabilistic theory of lattice models, using Bernoulli percolation and the Random-Cluster model as our main examples. We then use these tools to discuss some very recent results on the study of random Lipschitz functions.

A particular focus will be given to the 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 a presentation of Smirnov's proof of the conformal invariance of critical site percolation on the triangular lattice.

These models give a beautiful way to apply the material learnt in the Probability course. Quite a few ideas are of combinatorial nature and the field is connected to several other branches of mathematics: Mathematical Physics, Ergodic Theory, Complex Analysis, Conformal Geometry, Computer Science.

It is highly recommended to follow the exercise sessions.