Second meeting
15-16 October 2026

Giovanni Peccati: Filamentary structures in Gaussian random waves

Abstract: The term Berry’s random wave describes a canonical model for a Gaussian Laplace eigenfunction on the plane. This universal object naturally arises in connection with several questions in stochastic geometry, especially those concerning the local behaviour of Laplace eigenfunctions (random or deterministic) on chaotic surfaces. After a brief introduction to this model, I will highlight one striking (and still largely mysterious) structure that emerges in high-frequency regimes: scale-free, randomly scattered chains of narrow excursions, sometimes referred to as scars or scarlets, that appear in simulations of Berry’s random wave over large domains. In my talk, I will relate these structures to a notion of universality class involving a remarkable family of fractional Gaussian fields, naturally represented in terms of white noise on the Grassmannian of lines. All CLTs described in the talk are obtained through a combination of Stein’s method and Malliavin calculus. Based on joint work with L. Gass and M. Stecconi (Luxembourg).

Guillaume Poly: Martingale methods for Breuer -Major Theorem 

Abstract: The Breuer-Major CLT is a classic result in stochastic analysis which generalizes the standard CLT and has attracted a lot of attention in the past decades under the impulsion of Wiener chaoses related techniques. In this talk we will focus more precisely on the infinite dimensional counterpart which generalizes the classic Donsker CLT and will provide new conditions which generalize some previous seminal results of D.Nualart and I.Nourdin. A notable feature of our approach combines martingale techniques with hypercontractivity properties of the Ornstein-Uhlenbeck semi-group. At the end of talk, we shall discuss some open questions around this question. This is based on a joint work with G.Zheng (Boston-university) and P.Mansanarez (ULB and Nantes university).

Tara Trauthwein: Quantitative Central Limit Theorems for Poisson Point Processes via Localization

Abstract: Consider a Poisson point process on a metric space and a function thereof which consist of a sum of 'local' components. Examples can be statistics of spatial random graphs (total edge length, number of isolated points,...), or statistics of more complex models like the birth-growth model, which is an interacting particle system. It is known that when those functions exhibit a certain kind of 'local' behaviour, then they satisfy a Central Limit Theorem when the underlying point process grows. In this talk, we will explore what 'local' precisely means, and how Stein's method, combined with Malliavin calculus can lead to a way of showing Berry-Esseen type convergence rates. I will talk about a joint work with Joseph Yukich, where we propose the notion of ‘localization’, a weak condition on score functions which is sufficient to induce a CLT. We give Berry-Esseen type quantitative convergence rates and illustrate our results on several applications, among them the birth-growth model. The proof builds on (and improves) the Malliavin-Stein method.

Pierre Monmarché: Local convergence for mean field particle systems

Abstract: Motivated by many recent algorithms which are either designed as mean-field interacting particle systems or can be retrospectively interpreted as such, the question of establishing quantitative long-time estimates for these processes have received much interest over recent years. However, when the mean-field limit admits several stationary solutions (or possibly periodic orbits), particles are only ergodic at a time-scale which is out of reach for simulations. Classical tools for ergodic Markov processes can then only provide non-informative results. We will present a point of view to get relevant bounds on extremely large time-scales in this situation, and review some recent results in this direction.

 Louis-Pierre Chaintron: ResNets of All Shapes and Sizes: Quantitative Large-Scale Theory of Training Dynamics

Abstract: We study the joint large-depth, large-width, and large-embedding limit of residual neural networks in the maximal local-update regime. For ResNets with two-layer perceptron blocks of depth (L), hidden width (M), and embedding dimension (D), we first establish a quantitative universality result: after a bounded number of training steps, the finite-network dynamics are approximated with error [O!\left(\frac1L+\frac{\sqrt D}{\sqrt{LM}}+\frac1{\sqrt D}\right)] with experiments suggesting that this scaling is tight during early training.  For Gaussian initialization, the limiting dynamics simplify considerably: the infinite-size ResNet is described by a Neural Mean ODE in a Hilbert space, with initial weights replaced by isonormal Gaussian maps, and gradient descent on the finite network converges to gradient descent on this limiting model. From a probabilistic viewpoint, the large-(D) limit is a mean-field limit over embedding coordinates, while the general analysis can be viewed as a quantitative and rigorous counterpart of dynamical mean field theory. Our proofs combine propagation-of-chaos ideas with a functional cavity method. Together, these results provide both quantitative finite-size scaling laws and an optimization-preserving description of large-scale ResNet training. This is joint work with Lénaïc Chizat and Javier Maass..

Taiyang Xu: Large gap asymptotics for the confluent hypergeometric kernel determinant

Abstract: The confluent hypergeometric point process represents a universality class which arises in a variety of different but related areas. It particularly describes the local statistics of eigenvalues in the bulk of the spectrum near a Fisher–Hartwig singular point for a broad class of unitary ensembles. In this talk, I will first briefly review recent developments in the large gap asymptotics of some classical universal point processes, including the sine, Airy and Bessel point processes. I will then present the large gap asymptotics of the confluent hypergeometric point process on several disjoint intervals, focusing on the case where the Fisher–Hartwig singularity lies inside one of the intervals. A particularly interesting feature of the resulting asymptotic formula is its connection with a linear flow on a torus. At the end of this talk, I will also briefly discuss the richer asymptotic phenomena that arise when the Fisher–Hartwig singularity lies outside the intervals, including different regimes in which gaps merge at the singular point. This talk is based on several joint works with Lun Zhang and Zhengyang Zhao (Fudan University).

 Joakim Cronvall: A direct approach to soft and hard edge universality for random normal matrices

Abstract: The random normal matrix model is a two-dimensional point process describing log-correlated particles in a confining field. The statistics are described by the reproducing kernel of a weighted polynomial Bergman space. Typically, one studies the kernel using asymptotics of orthogonal polynomials. In many interesting situations, such as when the eigenvalues are confined by a hard wall or when the eigenvalues concentrate on disconnected sets, this has turned out to be complicated. In this talk, I will discuss an alternative approach that altogether avoids orthogonal polynomials. Instead we work with Hilbert spaces of entire functions using Paley-Wiener type theorems and potential theory.  This approach gives several new universality results and shows in particular how the microscopic behavior of the kernel depends on the local potential theory. Based on joint work with Aron Wennman