53th Probability Summer School
Saint-Flour (France), June 30th - July 12th 2025
Lectures 2025
Justin SALEZ
(Université Paris Dauphine CEREMADE)
Modern aspects of Markov chains: entropy, curvature, and the cutoff phenomenon.
The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to its maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to interacting particle systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. In this course, I will provide a self-contained introduction to this fascinating question and explore its recently discovered relations with the fundamental concepts of entropy, curvature and concentration.
Peter BARTLETT
(University of California at Berkeley, Berkeley AI Research Lab)
Deep learning: a statistical perspective.
Deep learning, the technology underlying the recent progress in AI, has revealed some major surprises from the perspective of theory. These methods seem to achieve their outstanding performance through different mechanisms from those studied in classical learning theory, mathematical statistics, and optimization theory. Simple gradient methods find excellent solutions to non-convex optimization problems, and without any explicit effort to control model complexity, they exhibit excellent prediction performance in practice. This lecture series will review recent progress on deep learning, viewed as a nonparametric statistical methodology, as well as some of the intriguing questions that it raises.
Massimiliano GUBINELLI
(University of Oxford, Mathematical Institute)
A stochastic analysis perspective on Euclidean fields.
The goal of these lectures is to introduce Euclidean quantum field theory as natural objects in probability theory: higher dimensional generalisations of Markov processes. I will explain the motivations and problems inherent in a rigorous theory of these objects and indicate how it is possible to use the fundamental ideas of stochastic analysis to tackle them, at least in dimensions two and three. We will explore various ways to construct Euclidean fields by solving certain stochastic analysis problems, a method called stochastic quantisation. If time permits I will also touch upon the open problem of giving a mathematical characterisation of Euclidean fields.