The Aztec diamond is one of the central exactly solvable models in probability and combinatorics. Its striking geometric features—most notably limit shapes and arctic boundaries—have made it a cornerstone of integrable probability. Over the past decade, weighted variants with doubly periodic edge weights have shown that much of this integrable structure persists far beyond the uniform case. These extensions have revealed new phenomena, including regions of smooth disorder. At the heart of these developments lies a birational transformation that encodes the integrable character of the system. In this talk, I will present recent joint work with Roger van Peski on a new disordered version of the Aztec diamond obtained by assigning independent Gamma-distributed weights to its edges. Despite the introduction of randomness at the level of weights, the model retains enough algebraic structure to remain integrable. This allows us to rigorously verify new behaviors that arise as a consequence of the disorder, such as $n^{2/3}$-scale fluctuations near the turning points of the arctic boundary. Strikingly, these turning points are closely related to certain integrable polymer models.

Je présenterai quelques aspects d'un lien découvert récemment reliant l'étude en temps long de systèmes intégrables hamiltoniens à celle des Beta-Ensembles des matrices aléatoires ; via l'étude des propriétés spectrales de certaines matrices aléatoires (appelées matrices de Lax). Je présenterai ces objets et quelques résultats en me concentrant sur l'exemple de la chaîne de Toda, reliée à l'étude du Beta-Ensemble réel.

On se donne un champ aléatoire $F(x,\omega)$ et une variable aléatoire $\Theta(\omega)$. Considérons ensuite des approximations $F^N$ de $F$ et $\Theta^N$ de $\Theta, pour un paramètre de convergence $N \to +\infty$.

Les questions naturelles sont les suivantes : a-t-on convergence de la composée $F^N(\Theta^N(\omega), \omega)$ vers $F(\Theta(\omega), \omega)$?

À quelle vitesse cette convergence a-t-elle lieu, et comment dépend-elle des vitesses de convergence de $F^N$ vers $F$ et de $\Theta^N$ vers $\Theta ?

Je présenterai ensuite quelques applications de ces résultats, notamment à la résolution de certaines EDP de type HJB, ainsi qu’à des problèmes de réduction de mémoire.

Consider the (2+1)D Discrete Gaussian (integer-valued Gaussian free field) on an $L\times L$ box above a floor. Bricmont, El-Mellouki and Fröhlich (1986) proved at low temperature there is entropic repulsion: the floor propels the average height to be poly-logarithmic in $L$. A more detailed picture was established by Lubetzky, Martinelli, Sly (2016), showing the height function looks like a plateau.

We answer some conjectures about the boundary of this plateau, i.e. the top level-line, as $L$ diverges. The global scaling limit is either the full box or a translation of Wulff shapes, depending on the sequence of $L$. In either case, along the flat sides of the limit shape, the top level-line converges to a Ferrari–Spohn diffusion on intervals of length $N^{2/3}$ after scaling by $(N^{2/3}, N^{1/3})$, for an explicit $N = L^{1-o(1)}$. In particular, for ``most'' choices of $L$, the fluctuations are $o(L^{1/3})$. Moreover, the joint law of any finite number of top level-lines, appropriately rescaled, is independent Ferrari–Spohn diffusions. These new results extend to the full universality class of grad-phi models for any fixed p>1. Joint work with Eyal Lubetzky.

K-means is a very well-known and widely used algorithm for clustering Euclidean data. It can be generalized to metric measure spaces, however, it is less well understood in settings where both the distance and the measure are unknown and must be estimated. In this talk, I will present recent work in which we prove the consistency of k-means in this context by establishing the stability of k-means centroids and clusters with respect to the measured Gromov–Hausdorff topology. This framework provides a unified approach for proving consistency across a wide range of metric learning procedures. In particular, I will discuss consequences for Isomap and Fermat geodesic distances on manifolds, diffusion distances, and Wasserstein distances. Work in collaboration with P. Groisman (Buenos Aires), M. Jonckheere (Toulouse) and M. Sued (Buenos Aires).

This work addresses the identifiability of First-order Vector AutoRegressive (VAR(1)) models in a stationary setting where time-series observations are unavailable, a common challenge in ecological research limited to independent samples from a stationary distribution. Recovering the dynamic interaction graph from static data is non-trivial because the autoregressive interaction graph does not coincide with the graphical model of the steady-state distribution, rendering standard Gaussian Graphical Model tools inapplicable. To resolve this, we adopt an approach from algebraic statistics, using Jacobian matroids of the model’s parametrization map to derive sufficient graphical conditions under which distinct dynamic networks yield distinguishable stationary distributions. This offers a framework for reconstructing dynamical interactions solely from equilibrium data.

TBA

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

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.