Mean-field spin glass models are statistical physics models where a large number of units interact through a random energy function with many symmetries. Their rich phenomenology makes them ideal for developing tools and ideas aimed at tackling high-dimensional optimization problems. For some of the models relevant to those applications, the units are partitioned into several species.

For many spin glass models, it is known that the free energy can be expressed as the maximal value of an explicit functional; this is the celebrated Parisi formula. Moreover, for single-species models, it is also known that the variational problem appearing in the Parisi formula admits a unique optimizer. This allows one to derive very precise properties of the energy landscape using the Parisi formula.

In this talk, I will explain how those results can be adapted to the setting of multi-species models and the different ideas required to prove them. Surprisingly, this involves studying a Hamilton-Jacobi partial differential equation. This is joint work with Hong-Bin Chen and Jean-Christophe Mourrat.

We address the problem of estimating the drift parameter in a system of $N$ interacting particles driven by additive fractional Brownian motion of Hurst index \( H \geq 1/2 \). Considering continuous observation of the interacting particles over a fixed interval \([0, T]\), we examine the asymptotic regime as \( N \to \infty \). Our main tool is a random variable reminiscent of the least squares estimator but unobservable due to its reliance on the Skorohod integral. We demonstrate that this object is consistent and asymptotically normal by establishing a quantitative propagation of chaos for Malliavin derivatives, which holds for any \( H \in (0,1) \). Leveraging a connection between the divergence integral and the Young integral, we construct computable estimators of the drift parameter. These estimators are shown to be consistent and asymptotically Gaussian. Finally, a numerical study highlights the strong performance of the proposed estimators. The talk is based on a joint work with I. Nourdin and R. Shevchenko

On étudie un processus d’apprentissage par renforcement, inspiré par les fourmis, qui communiquent à l'aide de phéromones. On considère un graphe G, avec deux nids (des sommets, N1 et N2) et une source de nourriture (un autre sommet, F). À chaque étape, une fourmi: 1) part d’un nid (aléatoire, N1 ou N2) et réalise une marche aléatoire (pondérée par les poids des arêtes) jusqu’à une source de nourriture F; puis 2) à son retour, elle dépose des phéromones, c'est-à-dire renforce les arêtes (en ajoutant 1 à leur poids) appartenant au chemin aller auquel on a enlevé les boucles inutiles.

Pour des raisons techniques, nous étudions le cas des graphes séries-parallèles en triangle, c'est-à-dire des graphes obtenus en remplaçant chaque arête du triangle N1 N2 F par un graphe série-parallèle. On montre que les poids des arêtes (normalisés) convergent, vers des variables aléatoires nulles si les arêtes associées n’appartiennent pas à un plus court chemin d’un sommet de {N1 , N2 , F} à un autre.

Nous présenterons plusieurs outils utiles pour prouver cette convergence, notamment la comparaison avec des processus d'urnes, des techniques d'approximations stochastiques et des arguments combinatoires.

La présentation se basera sur un travail en commun avec Cécile Mailler

When and how can an attention mechanism learn to selectively attend to informative tokens, thereby enabling detection of weak, rare, and sparsely located features? We address these questions theoretically in a sparse-token classification model in which positive samples embed a weak signal vector in a randomly chosen subset of tokens, whereas negative samples are pure noise. In the long-sequence limit, we show that a simple single-layer attention classifier can in principle achieve vanishing test error when the signal strength grows only logarithmically in the sequence length L, whereas linear classifiers require sqrt(L) scaling. Moving from representational power to learnability, we study training at finite in a high-dimensional regime, where sample size and embedding dimension grow proportionally. We prove that just two gradient updates suffice for the query weight vector of the attention classifier to acquire a nontrivial alignment with the hidden signal, inducing an attention map that selectively amplifies informative tokens. We further derive an exact asymptotic expression for the test error and training loss of the trained attention-based classifier, and quantify its capacity – the largest dataset size that is typically perfectly separable – thereby explaining the advantage of adaptive token selection over nonadaptive linear baselines. Joint work with Nicholas Barnfield and Yue M Lu.

Studying large scale dynamics of interacting many-particle systems is extremely demanding both analytical and numerical. Fortunately, there exist an effective asymptotic description, called hydrodynamics, based on averaging over local equilibrium states. Unfortunately, in general it is an open problem to understand its emergence, i.e. to connect the microscopic and the macroscopic description.

This motivates to study the problem of hydrodynamic emergence in simple models with exact microscopic solutions, such as the hard rods model (hard spheres in 1D). In arXiv:2408.04502, we had derived the diffusive-scale hydrodynamic equation. To our surprise the result was not of the expected Navier-Stokes type.

To investigate this, we propose a radical strategy: to study the evolution of a single deterministic configuration, instead of a statistical ensemble of them. First, I will explain how one can meaningfully define hydrodynamics in this scenario. Then, I will use the new to show that diffusion is in fact absent in hard rods and that hydrodynamics even works for “unphysical” configurations very far from local equilibrium. In addition, the method also provides error bounds on the quality of the hydrodynamic approximation.

Based on arXiv:2507.17827