# 52th Probability Summer School

Saint-Flour (France), 1 - 13 July 2024

## Lectures 2024

**Jean-Christophe Mourrat**

(ENS Lyon, France)

*Statistical mechanics of mean-field disordered systems.*

The goal of statistical mechanics is to describe the large-scale behavior of collections of simple elements, often called spins, that interact through locally simple rules and are influenced by some amount of noise. We will mostly be interested in the case when the local interactions themselves are chosen randomly, leading to models called "spin glasses". In the simplest case, there are interactions between any two units in the system. For modelling purposes, it is also desirable to consider models with more structure, such as when the units are split into two groups, and the interactions only go from one group to the other one. The course is meant as an introduction to some rigorous results on this topic, with an emphasis on the broader class of models.

**Sylvia Serfaty**

(Courant Institute of Mathematical Sciences New York University)

*Systems with Coulomb Interactions : Mean-field Limits and Statistical Mechanics.*

We will discuss large systems of particles with Coulomb-type repulsion. The first part of the course will mention the question of mean-field for the dynamics of such systems via a modulated energy approach. The second part concerns the statistical mechanics of such systems (expansion of free energy, LDP for empirical field, fluctuations around the mean-field limit). Several tools and ideas are common to both parts.

**Christina Goldschmidt**

(University of Oxford)

*Scaling limits of random trees and critical random graphs.*

Over the last 30 years or so, a beautiful theory of scaling limits for random trees has been established which, for example, encompasses many natural combinatorial models, as well as conditioned branching process trees. I will begin this course by giving an account of some of this theory, with a particular emphasis on the importance of algorithms for generating random trees in analysing them. I will discuss the Brownian continuum random tree and other stable trees, which are some of the key the random fractal structures arising as scaling limits in this context.

I will then show how this theory may be built upon in order to prove scaling limits in the context of random graphs. The prototypical model in this field is the Erdos-Rényi random graph G(n,p), in which there are n labelled vertices, each pair of which is independently joined by an edge with probability p. We will consider the case where p = c/n for some constant c > 0, so that the average degree of a vertex is roughly c. The G(n,p) model undergoes a phase transition in the sense that, for c < 1, the connected components are all microscopic (in the sense that each contains only a vanishing proportion of the vertices), whereas for c > 1, there is a single component (“the giant”) containing a positive proportion of the vertices, and the others are again all microscopic. As is often the case, the most delicate behaviour occurs at the point of the phase transition, and this critical regime will be our focus. For c = 1, it turns out that the largest components have sizes on the intermediate scale of n^{2/3} and are tree-like, in the sense that they have a tight number of edges more than a tree. Extending the scaling limit theory from trees to these graph components is the key to proving a scaling limit for the critical random graph. Moreover, similar techniques may be deployed in great generality on other models which undergo such a phase transition; I will, in particular, give an account of what happens for the configuration model with independent and identically distributed random vertex degrees, both when this graph is in the same universality class as G(n,p), and when the vertex-degrees have heavier tails.

In the final part of the course, I hope to explore some of the implications of the earlier material, including the analysis of models in which the edges are directed, as well as a proof of the existence of a scaling limit for the minimum spanning tree of the complete graph, which turns out to be intimately related to the scaling limit of G(n,p).