Danish Maritime Authority

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.

In this paper, we explore the statistical subtleties of the nonideal Rayleigh gas, in a grand canonical mixture framework. This model allows to consider a large amount of tagged particles close to equilibrium, and their empirical measure, whose first-order convergence has been shown to converge to the solution of the linear Rayleigh-Boltzmann equation [5]. Thanks to the study of the cumulants of the system, we analyze the asymptotic behaviour of the fluctuations and large deviations of this empirical measure, hence refining the previous statistical results in the same vein as [7]. This way, we exhibit the trivial limit behaviour of the fluctuations in any overdilute regime, proving the exact relevance at any statistical scale of the low density limit. In the case of large deviations, we present the linear Boltzmann-Hamilton-Jacobi system driving their asymptotic behaviour. Eventually, we optimize the geometrical estimates on the billiards dynamics [6] to finally achieve a full convergence rate for the cumulants.

This paper introduces a grand canonical mixture model to generalize the nonideal Rayleigh gas [5] to an asymptotically infinite amount of perturbed tagged particles. This model relies precisely on grand canonical tags, to preserve symmetry in the system, contrary to [2]. We hence define and study the convergence of the correlation functions of this system in large times, linking it to the expectancy of the empirical measure of tagged and non-tagged particles, to eventually prove a law of large numbers for this dynamics. We extend the quantitative study to all the correlation functions, and not only the first one, exhibiting the resultant additional factors, and we also generalize the perturbation to the whole phase space, instead of considering a space-only initial perturbation. Eventually, we fit our adaptive time cutting [12] to the mixture system, even improving it to get better convergence rates.

We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein-Uhlenbeck process as well as non Gaussian and non product extensions with convex interaction, such as the Dyson process from random matrix theory. We show that a cutoff phenomenon or abrupt convergence to equilibrium occurs in high-dimension, at a critical time equal to the logarithm of the dimension divided by twice the spectral gap. This cutoff holds for Wasserstein distance, total variation, relative entropy, and Fisher information. A key observation is a relation to a spectral rigidity, linked to the presence of a Gaussian factor. A novelty is the extensive usage of functional inequalities, even for short-time regularization, and the reduction to Wasserstein. The proofs are short and conceptual. Since the product condition is satisfied, an Lp cutoff holds for all p. We moreover discuss a natural extension to Riemannian manifolds, a link with logarithmic gradient estimates in short-time for the heat kernel, and ask about stability by perturbation. Finally, beyond rigidity but still for diffusions, a discussion around the recent progresses on the product condition for nonnegatively curved diffusions leads us to introduce a new curvature product condition.

In this expository note, we return to the classification of exceptional actions of PSL2(Z/pZ) on a set with p elements, announced by Galois in his last letter to Chevalier. Mathematics Subject Classification 2020: 20G40 (primary); 05E18 (secondary).