54th Probability Summer School
Saint-Flour (France), June 29th - July 11th 2026

This course aims to cover two topics related to the analysis of large systems of weakly interacting particles and, ultimately, to bring them together. The common foundation of the two directions that will be pursued is the theory of diffusive mean-field models, or McKean–Vlasov stochastic differential equations. Intuitively, these models describe the asymptotic statistical evolution of large exchangeable populations, subject to independent noise and interacting with one another through the marginal empirical measure of the system. In the limit, the particles become independent, so that the marginal empirical measure converges toward a deterministic flow of probability distributions.

Schematically, we will study the following two directions:
(i) on the one hand, we will consider the case in which the particle dynamics are not prescribed a priori, but instead result from an optimization procedure; in particular, we will see how the optimization step interacts with the mean-field limit;
(ii) on the other hand, we will pay special attention to mean-field models in which the limiting dynamics, viewed as a system taking values in the space of probability measures, are randomized; the objective is to extract from this randomization additional exploration properties of the space of probability measures.

The two topics are connected as follows: for (i), we will show that, in the mean-field regime, the optimization problem can be characterized by a nonlinear partial differential equation posed on the space of probability measures; for (ii), we will investigate which regularization properties the mean-field semigroup—acting on functions of a probability measure—inherits from the randomization. We will then combine (i) and (ii) in order to exploit randomization to solve nonlinear equations posed on the space of probability measures.

Direction (i) is connected to the theory of mean-field games and mean-field control, which has enjoyed great success in probability theory, PDEs, and the calculus of variations since the courses of P.-L. Lions at the Collège de France on the subject. Direction (ii) is related to the notion of Wasserstein diffusion; in particular, the course will include an overview of the different types of processes that have been developed in this framework and will address the link with the Dean–Kawasaki equation.

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Planar maps are graphs embedded in the sphere such that no two edges cross, where we view two planar maps as equivalent if we can get one from the other via a continuous deformation of the sphere. Planar maps are studied in several different branches of mathematics and physics. In particular, in probability theory and theoretical physics random planar maps are used as natural models for discrete random surfaces. In this course we will present scaling limit results for random planar maps and we will focus in particular on a notion of convergence known as convergence under conformal embedding. The limiting surface is a highly fractal surface called a Liouville quantum gravity (LQG) surface, which has its origin in string theory and conformal field theory.

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In Bernoulli bond percolation, each edge of a graph is either deleted or retained independently at random with retention probability p. On many large finite graphs, percolation has a phase transition, meaning that a giant (i.e. positive density) cluster (i.e. connected component) emerges as p is varied through a critical value. This course will examine the basic qualitative properties of this giant component in the setting of transitive graphs, i.e., graphs for which all vertices "look the same". After reviewing the classical theory of uniqueness and non-uniqueness of the infinite cluster for percolation on infinite transitive graphs, I will overview recent developments in the qualitative theory of the giant component for percolation on large finite transitive graphs. In particular, I will give a full proof of a recent result, joint with Easo, showing that the giant cluster is unique in the supercritical regime in essentially all large finite transitive graphs aside from an explicit family of high-degree counterexamples we call "molecular graphs". Time permitting, I may also overview some other recent related developments in the infinite graph setting.

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