Abstract. The course will focus on rigorous results for the self-avoiding walk model on lattices, with a special emphasis on low-dimensional ones. The model is defined by choosing uniformly at random among random walk paths starting from the origin and without self-intersections. Despite its simple definition, the self-avoiding walk is difficult to comprehend in a mathematically rigorous fashion, and many of the most important problems illustrating standard challenges of critical phenomena remain unsolved. The model is combinatorial in nature but many questions about the stochastic properties of these random paths can be solved by combining nice combinatorial features with probabilistic techniques. In the course, we will describe some of the recent techniques developed in the area, including the use of discrete holomorphicity to understand the model on the hexagonal lattice.

Abstract. Ce cours s'intéresse à la transition de phase du modèle d'Ising sur le réseau Zd et à son comportement critique. Nous montrerons qu'en dimensions quatre et plus, le modèle se comporte comme en champ moyen. Nous prouverons également l'invariance conforme du modèle en dimension deux. Enfin, nous complèterons cette étude par quelques résultats partiels et questions ouvertes sur le modèle d'Ising en dimension trois. L'étude combine des arguments géométriques avec une représentation, appelée représentation en courants aléatoires, introduite dans les années quatre-vingt par Michael Aizenman.

Abstract. In this course, we will present different techniques developed over the past few years, enabling mathematicians to prove that phase transitions are sharp. We will focus on a few classical models of statistical physics, including Bernoulli percolation, the Ising model and the random-cluster model.

Abstract. Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by $\sqrt{2+\sqrt 2}$, and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.