Plenary Lecture or Open Public Lecture

The cutoff phenomenon for Markov chains

Justin Salez

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. 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 talk, I will provide a self-contained introduction to this fascinating question, and then describe a recent partial answer based on entropy and curvature. Joint work with Francesco Pedrotti.

On rates in the central limit theorem for a class of convex costs

Florence Merlevède

In this talk, we will shall give estimates not only of the usual quadratic transportation cost, but also of a broader class of convex costs between normalized partial sums associated with real-valued random variables and their limiting Gaussian distribution. For the quadratic transport cost, estimates will be given in terms of weak-dependent coefficients that are well suited to a large class of dependent sequences. This class includes irreducible Markov chains, dynamical systems generated by intermittent maps or strong mixing sequences. We will also present very recent results in the independent framework for a broader class of convex costs that includes quadratic cost as a special case but also certain logarithmic costs such as \(|x| \ln (1+ |x|)\). This talk is based on joint works with J. Dedecker and E. Rio.

Critical long-range percolation

Tom Hutchcroft

Many statistical mechanics models on the lattice (including percolation, self-avoiding walk, the Ising model and so on) have natural "long-range" versions in which vertices interact not only with their neighbours, but with all other vertices in a way that decays with the distance. When this decay is described by a power-law, it can lead to new kinds of critical phenomena that are not present in the short-range models. I will describe a new approach to the study of these models that allows us to go well beyond what is known for short-range models, particularly in intermediate dimensions.

On estimating Fr'{e}Chet means

Roberto Imbuzeiro Oliveira

A Fréchet mean (or barycenter) of a distribution \(P\) over a metric space \((\mathcal{X},d)\) is any point \[\mu\in {\rm arg}\min_{m\in\mathcal{X}}\mathbb{E}_{X\sim P}\,d^2(X,m).\] This definition makes sense for any \((\mathcal{X},d)\), and coincides with the expectation of \(P\) if \(\mathcal{X}\) is Euclidean space. This talk considers the problem of estimating Fréchet means from an i.i.d. sample from \(P\), possibly contaminated by adversarial noise. The goal is to obtain optimal or-nearly optimal high probability bounds under weak moment assumptions on \(P\).

Two main settings are considered. The first one consists of Alexandrov spaces where geodesics are “bi-extendible” in some precise quantitative sense, which is rich enough to include the space of probability measures over \(\mathbb{R}^d\) with the Wasserstein-\(2\) distance. Our main theorem for this case refines an analysis of \([1]\), and obtains bounds with the “right” variance parameters The second setting we consider is that of uniformly convex Banach spaces of power type \(2\), where our results seem to be completely new. Both results are obtained via an analysis of certain “trimmed empirical processes”  that is based on previous work with Lucas Resende \([2]\).

Bibliography

\([1]\) T. Le Goı̈c, Q. Paris, P. Rigollet and A. Stromme. “Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space.” Journal of the European Mathematical Society (2022), Vol 25(6), 2229–2250.
\([2]\) R. I. Oliveira and L. Resende. “Trimmed sample means for robust uniform mean estimation and regression.” arXiv:2302.06710.

What AI will not tell you about white noise

Krzysztof Burdzy

This lecture is a brave attempt to show that human intelligence might still be useful in the era of artificial intelligence. White noise and Brownian motion are closely related fundamental concepts in science and mathematics. Their attraction stems in part from their paradoxical mixture of simplicity and complexity.

The lecture will be illustrated and accessible to a wide audience.

Exchangeability in Continuum Random Trees

Minmin Wang

De Fenetti’s Theorem states that all \(\mathbb N\)-indexed exchangeable sequences of real-valued random variables are mixings of i.i.d sequences. For real-valued random processes with exchangeable increments on \([0, 1]\), Kallenberg’s result \([1]\) provides a complete characterisation of these processes via another mixing relationship.

Continuum random trees are random tree-like metric spaces that arise naturally as scaling limits of various models of discrete random trees. In this talk, we will focus in particular on two subclasses of continuum random trees: the so-called stable trees and inhomogeneous continuum random trees. An analogue of Kallenberg’s Theorem for continuum random trees first appeared as a claim in a paper \([2]\) by Aldous, Miermont and Pitman. They suggested that, in much the same way that a stable bridge process on \([0, 1]\) is a mixing of certain extremal exchangeable processes, stable trees are mixings of inhomogeneous continuum random trees.

We present an outline of a rigorous argument supporting this claim, based on a novel construction that applies to both classes of trees. We will also briefly discuss some implications of this result on critical random graphs.

The talk is based on the paper \([3]\).

Bibliography

\([1]\) O. Kallenberg. “Canonical representations and convergence criteria for processes with interchangeable increments.” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 27, 1973, pp. 23–36.

\([2]\) D. Aldous, G. Miermont and J. Pitman. “The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity." Probab. Theory Related Fields, vol. 129, no. 2, 2004, pp. 182–218.

\([3]\) M. Wang. “Stables trees as mixings of inhomogeneous continuum random trees.” Stochastic Process. Appl., vol. 175, 2024.

Bootstrap percolation and kinetically constrained models: universality results

Toninelli Cristina

Recent years have witnessed significant progress in the study of bootstrap percolation (BP) models. In the initial configuration sites are occupied with probability p. The evolution of BP proceeds in discrete time: empty sites remain empty, while occupied sites become empty if and only if a certain model-dependent neighborhood is already empty. On Z^d there is now a fairly complete understanding of the dynamics starting from random initial conditions, along with a clear universality picture for their critical behavior. Much less is known about their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In these models each vertex is either infected or healthy and, iff it is infectable according to the BP rules, its state is resampled (independently) at rate one and becomes infected with probability p, and healthy with probability 1-p. These models, introduced and intensively studied in physics literature as toy models of the liquid/glass transition, present both challenging and fascinating mathematical problems. Indeed, the presence of constraints induce non-attractiveness, multiple invariant measures, and the breakdown of many powerful tools (such as coercive inequalities, coupling arguments, and censoring techniques) typically used to study convergence to equilibrium. In this talk, I will present a series of results that establish the full universality picture of KCM in two dimensions. We will see that, compared to those of BP, the universality classes for the stochastic dynamics are richer and the critical time scales diverge more rapidly due to the dominant role of energy barriers. The seminar is based on joint works with I.Hartarsky, L.Marêché, F.Martinelli, and R.Morris.