LFU:online

Students who have completed that module have acquired particular knowledge of one or more branches of higher mathematics. They are able to develop innovative solutions for current problems of those branches of mathematics as well as to judge different approaches critically.
As a result they have developed learning strategies that enable them to acquire further mathematical matters autonomously.

Our goal is to introduce the audience to a beatiful and very active area of research in Probability Theory that studies Lattice models. These models have simple definitions but allow to construct deep mathematical theories. The growing interest to lattice models is confirmed with Fields Medals to Werner (2006), Smirnov (2010), Duminil-Copin (2022). Berezinskii–Kosterlitz–Thouless phase transition in the XY model was awarded Nobel Prize in Physics in 2016.

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.

Pre-requisites: bachelor course on Probability Theory.

No previous knowledge of Physics/Mathematical Physics is needed.

Continuous assessment (based on regular written and/or oral contribution by participants).

Oral examinations are examinations that require responding to questions verbally; according to § 6, statute section on "study-law regulations"

Lecture notes of Hugo Duminil-Copin:
- https://www.ihes.fr/~duminil/publi/2017percolation.pdf
- https://arxiv.org/pdf/1707.00520.pdf

Grimmett "The Random-cluster model":
- http://www.statslab.cam.ac.uk/~grg/books/rcm1-1.pdf

The course will be more or less evenly divided into lectures and exercise sessions. The subject is perfectly suited for learning by solving problems. This will also allow to have more interaction and to learn from other students.