Nos 50 dernières publications

8 April 2024 hal04536467
We introduce a numerical strategy to study the evolution of twodimensional water waves in the presence of a plunging jet. The freesurface Navier–Stokes solution is obtained with a finite, but small, viscosity. We observe the formation of a surface boundary layer where the vorticity is localised. We highlight convergence to the inviscid solution. The effects of dissipation on the development of a singularity at the tip of the wave is also investigated by characterising the vorticity boundary layer appearing near the interface.
Alan Riquier, Emmanuel Dormy

7 April 2024 hal04519638
In this article, we establish embeddings à la Lions and transfer of regularity à la Bouchut for a large scale of kinetic spaces. We use them to identify a notion of weak solutions to KolmogorovFokkerPlanck equations with (local or integral) diffusion and rough (measurable) coefficients under minimal requirements. We prove their existence and uniqueness for a large class of source terms, first in full space for the time, position and velocity variables and then for the kinetic Cauchy problem on infinite and finite time intervals.
Pascal Auscher, Cyril Imbert, Lukas Niebel

3 April 2024 hal04531083
This paper’s objective is to improve the existing proof of the derivation of the Rayleigh–Boltzmann equation from the nonideal Rayleigh gas [6], yielding a far faster convergence rate. This equation is a linear version of the Boltzmann equation, describing the behavior of a small fraction of tagged particles having been perturbed from thermodynamic equilibrium. This linear equation, derived from the microscopic Newton laws as suggested by the Hilbert’s sixth problem, is much better understood than the quadratic Boltzmann equation, and even enable results on long time scales for the kinetic description of gas dynamics. The present paper improves the physically poor convergence rate that had been previously proved, into a much more satisfactory rate which is more than exponentially better.
Florent Thomas Fougères

27 March 2024 hal04519657
We show the existence and uniqueness of fundamental solution operators to Kolmo\gorovFokkerPlanck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is to say when the diffusion operator is of differential type, we prove $\L^2$ decay using Davies' method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser's $\L^2\L^\infty$ estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay.
Pascal Auscher, Cyril Imbert, Lukas Niebel

26 March 2024 hal04521375
We identify the local limit of massive spanning forests on the complete graph. This generalizes a wellknown theorem of Grimmett on the local limit of uniform spanning trees on the complete graph.
Matteo d'Achille, Nathanaël Enriquez, Paul Melotti

11 March 2024 hal04293802
We study the Maximum ZeroSum Partition problem (or MZSP), defined as follows: given a multiset S={a_{1}, a_{2}, ..., a_{n}} of integers a_{i}∈Z* such that Σ_{i=1..n} a_{i}=0, find a maximum cardinality partition {S_{1}, S_{2}, ... , S_{k}} of S such that, for every 1≤ i ≤ k, Σ_{aj ∈ Si} a_{j}=0. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixedparameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S.
Guillaume Fertin, Oscar Fontaine, Géraldine Jean, Stéphane Vialette

7 March 2024 hal04494574
This paper aims at developing modeltheoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from generic data is proved. It relies heavily on a stabilizer theorem of Montenegro, Onshuus and Simon. The main application is a structure theorem for definably amenable groups that are interpretable in algebraically bounded perfect $\mathrm{NTP_2}$ fields with bounded Galois group (under some mild assumption on the imaginaries involved), or in algebraically bounded theories of (differential) NIP fields. These imply a classification of the fields interpretable in differentially closed valued fields, and structure theorems for fields interpretable in finitely ramified henselian valued fields of characteristic $0$, or in NIP algebraically bounded differential fields.
Paul Z. Wang

6 March 2024 hal04493074
We study one and twodimensional periodic tightbinding models under the presence of a potential that grows to infinity in one direction, hence preventing the particles to escape in this direction (the soft wall). We prove that a spectral flow appears in these corresponding edge models, as the wall is shifted. We identity this flow as a number of Bloch bands, and provide a lower bound for the number of edge states appearing in such models.
Camilo Gómez Araya, David Gontier, Hanne van Den Bosch

5 March 2024 hal04490266
The socalled Faster is Slower (FIS) effect is observed in some particular reallife or experimental situations. In the context of an evacuation process, it expresses that increasing the speed (or, more generally, the competitive ness) of individuals may induce a reduction of the flow through the exit door. We propose here a parameterfree model to reproduce and investigate this effect (more precisely its backward “Slower is Faster” equivalent). In spite of its nonsmooth character, which makes it difficult to analyze, this gran ular approach is based on very basic ingredients in terms of behavior. In its native, purely asocial version, individuals are represented by harddiscs, each of which has a desired velocity, and the actual velocity is built as the projection of this field on the set of admissible velocities (which respect the nonoverlapping constraints). We implement the slower effect by introducing here an extra step to account for the fact that individuals refrain from pushing, and therefore tend to reduce their desired velocity accounting for the velocities of people upfront. The present paper has two objectives: estab lish the relevance of this model by showing that it satisfactorily reproduces various empirical effects in highly crowded evacuations with various levels of competitiveness, and explore how it can be implemented to recover and explain the FIS effect. In this spirit, we confront this InhibitionBased (IB) model to experimental data, focusing on the Faster is Slower effect. We show in particular that this approach makes it possible to accurately recover the effect of competitiveness upon powerlaw distributions of time lapses which have been experimentally observed. We also study the effect of mixed behaviors, by introducing a twopopulation model using both approaches. Weinvestigate in particular the effect upon evacuation efficiency of the ratio be tween competitive agents and noncompetitive ones. In a similar context, we investigate the role of an obstacle placed upstream the exit upon evacuation efficiency.
Bertrand Antti Maury, Etienne Pinsard, Sylvain Faure, Fatima Al Reda

29 February 2024 hal04484591
This article focuses on a large family of crossdiffusion systems of the form ∂ t U∆A(U) = 0, in dimension d ∈ N * , and where U ∈ R 2. We show that under natural conditions on the nonlinearity A, those systems have a unique smooth (nonnegative for all components) solution when the initial data are small enough in a suitable norm.
L Desvillettes, A Moussa

20 February 2024 hal03344301
We consider conservative crossdiffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first control the time evolution of the gap between two bounded solutions by means of its initial value. As a by product, we obtain a uniqueness result for bounded solutions valid for any space dimension, under a nonperturbative smallness assumption. Using a discrete counterpart of our duality estimates, we prove the convergence of random walks with local repulsion in one dimensional discrete space to crossdiffusion systems. More precisely, we prove quantitative estimates for the gap between the stochastic process and the crossdiffusion system. We give first rough but general estimates; then we use the duality approach to obtain fine estimates under less general conditions.
Vincent Bansaye, Ayman Moussa, Felipe MuñozHernández

19 February 2024 hal04464740
We consider the nonlinear Schrödinger equation with double power nonlinearity. We extend the scattering result in [17] for all L 2supercritical powers, specially, our results adapt to the cases of energysupercritical nonlinearity.
Thomas Duyckaerts, Phan van Tin

17 February 2024 hal04463495
We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible. The classes that we consider are minimal cohomology classes on Jacobians of very general curves. Our main tool is complex cobordism.
Olivier Benoist, Olivier Debarre

17 February 2024 hal03807436
We show that any smooth cubic hypersurface of dimension n defined over a finite field Fq contains a line defined over Fq in each of the following cases: • n = 3 and q ≥ 11; • n = 4 and q 6= 3; • n ≥ 5. For a smooth cubic threefold X, the variety of lines contained in X is a smooth projective surface F(X) for which the Tate conjecture holds, and we obtain information about the Picard number of F(X) and the 5dimensional principally polarized Albanese variety A(F(X)
Olivier Debarre, Antonio Laface, Xavier Roulleau

17 February 2024 hal03807432
We show that any smooth cubic hypersurface of dimension n defined over a finite field Fq contains a line defined over Fq in each of the following cases: • n = 3 and q ≥ 11; • n = 4 and q 6= 3; • n ≥ 5. For a smooth cubic threefold X, the variety of lines contained in X is a smooth projective surface F(X) for which the Tate conjecture holds, and we obtain information about the Picard number of F(X) and the 5dimensional principally polarized Albanese variety A(F(X)
Olivier Debarre, Zhi Jiang, Martí Lahoz, William F Sawin

15 February 2024 hal03998535
We study a selfattractive random walk such that each trajectory of length $N$ is penalised by a factor proportional to $exp({R}_{N}\left\right)$, where ${R}_{N}$ is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately ${\rho}_{d}{N}^{1/(d+2)}$, for some explicit constant ${\rho}_{d}>0$. This proves a conjecture of Bolthausen [Bol94] who obtained this result in the case $d=2$.
Nathanael Berestycki, Raphaël Cerf

12 February 2024 hal04454075
Understanding the mechanisms behind the remote triggering of landslides by seismic waves at microstrain amplitude is essential for quantifying seismic hazards. Granular materials provide a relevant model system to investigate landslides within the unjamming transition framework, from solid to liquid states. Furthermore, recent laboratory experiments have revealed that ultrasoundinduced granular avalanches can be related to a reduction in the interparticle friction through shear acoustic lubrication of contacts. However, investigating slip at the scale of grain contacts within an optically opaque granular medium remains a challenging issue. Here, we propose an original coupling model and numerically investigate 2D dense granular flows triggered by basal acoustic waves. We model the triggering dynamics at two separated timescales—one for grain motion (milliseconds) and the other for ultrasound (10 microseconds)—relying the computation of vibrational modes with a discrete element method through the reduction of the local friction. We show that ultrasound predominantly propagates through the strongforce chains, while the ultrasoundinduced decrease of interparticle friction occurs in the weak contact forces perpendicular to the strongforce chains. This interparticlefriction reduction initiates local rearrangements at the grain scale that eventually lead to a continuous flow through a percolation process at the macroscopic scale—with a delay depending the proximity to the failure. Consitent with the experiment, we show that ultrasoundinduced flow appears more uniform in space than pure gravitydriven flow, indicating the role of an effective temperature by ultrasonic vibration.
Hugo A. Martin, Anne Mangeney, Xiaoping Jia, Bertrand Antti Maury, Aline LefebvreLepot, Yvon Maday, Paul Dérand

12 February 2024 hal02332759
Understanding the mechanisms behind the remote triggering of landslides by seismic waves at microstrain amplitude is essential for quantifying seismic hazards. Granular materials provide a relevant model system to investigate landslides within the unjamming transition framework, from solid to liquid states. Furthermore, recent laboratory experiments have revealed that ultrasoundinduced granular avalanches can be related to a reduction in the interparticle friction through shear acoustic lubrication of contacts. However, investigating slip at the scale of grain contacts within an optically opaque granular medium remains a challenging issue. Here, we propose an original coupling model and numerically investigate 2D dense granular flows triggered by basal acoustic waves. We model the triggering dynamics at two separated timescales—one for grain motion (milliseconds) and the other for ultrasound (10 microseconds)—relying the computation of vibrational modes with a discrete element method through the reduction of the local friction. We show that ultrasound predominantly propagates through the strongforce chains, while the ultrasoundinduced decrease of interparticle friction occurs in the weak contact forces perpendicular to the strongforce chains. This interparticlefriction reduction initiates local rearrangements at the grain scale that eventually lead to a continuous flow through a percolation process at the macroscopic scale—with a delay depending the proximity to the failure. Consitent with the experiment, we show that ultrasoundinduced flow appears more uniform in space than pure gravitydriven flow, indicating the role of an effective temperature by ultrasonic vibration.
Laurent Chevillard, Christophe Garban, Rémi Rhodes, Vincent Vargas

12 February 2024 hal01758101
We present a rigorous proof of the Dorn, Otto, Zamolodchikov, Zamolodchikov formula (the DOZZ formula) for the 3 point structure constants of Liouville Conformal Field Theory (LCFT) starting from a rigorous probabilistic construction of the functional integral defining LCFT given earlier by the authors and David. A crucial ingredient in our argument is a probabilistic derivation of the reflection relation in LCFT based on a refined tail analysis of Gaussian multiplicative chaos measures.
Antti Kupiainen, Rémi Rhodes, Vincent Vargas

10 February 2024 hal04398984
We consider a Schrödinger equation with a nonlinearity which is a general perturbation of a power" nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a general defocusing setting, assuming thatthe critical Sobolev norm is bounded in the energysupercritical case. This generalizes severalprevious works on doublepower nonlinearities.
Thomas Duyckaerts, Phan van Tin

6 February 2024 hal04441540
Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples and obstructions that prevent a similar result from holding on the halfsphere endowed with a uniform measure, answering a question of Beck and Jerison.
Max Fathi, Matthieu Fradelizi, Nathael Gozlan, Simon Zugmeyer

2 February 2024 hal04435902
Comparing probability distributions is a fundamental problem in data sciences. Simple norms and divergences such as the total variation and the relative entropy only compare densities in a pointwise manner and fail to capture the geometric nature of the problem. In sharp contrast, Maximum Mean Discrepancies (MMD) and Optimal Transport distances (OT) are two classes of distances between measures that take into account the geometry of the underlying space and metrize the convergence in law. This paper studies the Sinkhorn divergences, a family of geometric divergences that interpolates between MMD and OT. Relying on a new notion of geometric entropy, we provide theoretical guarantees for these divergences: positivity, convexity and metrization of the convergence in law. On the practical side, we detail a numerical scheme that enables the large scale application of these divergences for machine learning: on the GPU, gradients of the Sinkhorn loss can be computed for batches of a million samples.
Jean Feydy, Thibault Séjourné, FrançoisXavier Vialard, ShunIchi Amari, Alain Trouvé, Gabriel Peyré

2 February 2024 hal04435173
In this paper, we investigate the impact of stochasticity and large stepsizes on the implicit regularisation of gradient descent (GD) and stochastic gradient descent (SGD) over diagonal linear networks. We prove the convergence of GD and SGD with macroscopic stepsizes in an overparametrised regression setting and characterise their solutions through an implicit regularisation problem. Our crisp characterisation leads to qualitative insights about the impact of stochasticity and stepsizes on the recovered solution. Specifically, we show that large stepsizes consistently benefit SGD for sparse regression problems, while they can hinder the recovery of sparse solutions for GD. These effects are magnified for stepsizes in a tight window just below the divergence threshold, in the "edge of stability" regime. Our findings are supported by experimental results.
Mathieu Even, Scott Pesme, Suriya Gunasekar, Nicolas Flammarion

27 January 2024 hal01110901
We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3manifolds admitting a $\bS^1$symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from CMC results.
Romain Gicquaud, Cecile Huneau

24 January 2024 hal04413765
We study the parabolic defocusing stochastic quantization equation with both mutliplicative spatial white noise and an independant spacetime white noise forcing, on compact surfaces, with polynomial nonlinearity. After renormalizing the nonlinearity, we construct the random Gibbs measure as an absolutely continuous measure with respect to the law of the Anderson Gaussian Free Field for fixed realization of the spatial white noise. Then, when the initial data is distributed according to the Gibbs measure, we prove almost sure global wellposedness for the dynamics and invariance of the Gibbs measure.
Hugo Eulry, Antoine Mouzard, Tristan Robert

19 January 2024 hal04406829
We prove a local version of Gowers' Ramseytype theorem [25], as well as local versions both of the Banach space first dichotomy (the "unconditional/HI" dichotomy) of Gowers [25] and of the third dichotomy (the "minimal/tight" dichotomy) due to FerencziRosendal [22]. This means that we obtain versions of these dichotomies restricted to certain families of subspaces called Dfamilies, of which several concrete examples are given. As a main example, nonHilbertian spaces form Dfamilies; therefore versions of the above properties for nonHilbertian spaces appear in new Banach space dichotomies. As a consequence we obtain new information on the number of subspaces of nonHilbertian Banach spaces, making some progress towards the "ergodic" conjecture of FerencziRosendal and towards a question of Johnson.
Wilson Cuellar Carrera, Noé de Rancourt, Valentin Ferenczi

19 January 2024 hal04406804
We give new and simple proofs of some classical properties of hereditarily indecomposable Banach spaces, including the result by W. T. Gowers and B. Maurey that a hereditarily indecomposable Banach space cannot be isomorphic to a proper subspace of itself. These proofs do not make use of spectral theory and therefore, they work in real spaces as well as in complex spaces. We use our method to prove some new results. For example, we give a quantitative version of the latter result by Gowers and Maurey and deduce that Banach spaces that are isometric to all of their subspaces should have an unconditional basis with unconditional constant arbitrarily close to $1$. We also study the homotopy relation between into isomorphisms from hereditarily indecomposable spaces.
Noé de Rancourt

19 January 2024 hal04406796
We develop a general framework for infinitedimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers’ Ramseytype theorem for block sequences in Banach spaces and by its exact version proved by Rosendal. In this framework, we prove the adversarial Ramsey principle for Borel sets, a result conjectured by Rosendal that generalizes at the same time his version of Gowers’ theorem and Borel determinacy of games on integers.
Noé de Rancourt

11 January 2024 hal04388950
In this paper, we show how to extend the twin blowup method recently developped by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive HamiltonJacobi equation posed in a domain of an Euclidian space of any dimension and supplemented with a boundary condition. The method allows dealing with the case where tangential variables and the variable corresponding to the normal gradient of the solution are strongly coupled at the boundary. We elaborate on a method introduced by P.L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017). Their argument relies on a single blowup procedure after rescaling the semisolutions to be compared while two simultaneous blowups are performed in this work, one for each variable of the classical doubling variable technique. A onesided Lipschitz estimate satisfied by a combination of the two blowup limits plays a key role.
Nicolas Forcadel, Cyril Imbert, Regis Monneau

24 December 2023 hal04363393
We consider Markov processes with generator of the form $γ\mathcal{L}_{1} + \mathcal{L}_{0}$, in which $\mathcal{L}_{1}$ generates a socalled dominant process that converges at large times towards a random point in a fixed subset called the effective state space. Using the usual characterization through martingales problems, we give general conditions under which homogenization holds true: the original process converges, when $γ$ is large and for the MeyerZheng pseudopath topology and for finitedimensional time marginals, towards an identified effective Markov process on the effective space. Few simple model examples for diffusions are studied.
Dimitri Faure, Mathias Rousset

22 December 2023 hal04093949
Considering an integer d > 0, we show the existence of convexcocompact representations of surface groups into SO(4, 1) admitting an embedded minimal map with curvatures in (−1, 1) and whose associated hyperbolic 4manifolds are disk bundles of degree d over the surface, provided the genus g of the surface is large enough. We also show that we can realize these representations as complex variation of Hodge structures. This gives examples of quasicircles in S³ bounding superminimal disks in H⁴ of arbitrarily small second fundamental form. Those are examples of generalized almostFuchsian representations which are not deformations of Fuchsian representations.
Samuel Bronstein

22 December 2023 hal03984193
We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Zinvariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the twopoint spin correlations in the full plane behave as \[ \delta^{\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(u_{1}u_{2},m\right)\quad\text{as}\quad\delta\to0, \] where the universal constant $C_{\sigma}$ and the function $\Xi(u_{1}u_{2},m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multipoint spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the subcritical massive regime. We also give a simple proof of the fact that the infinitevolume magnetization in the Zinvariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{1/2}$ for $m=0$ and of $z^{1/2}e^{\pm 2mz}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.
Dmitry Chelkak, Konstantin Izyurov, Rémy Mahfouf

19 December 2023 hal04103354
Given a sequence of random variables $\left\{ X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions $$\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}$$ and $$\Omega_{\text{rand},\text{lac}} := \left\{ \frac{ \pi X_k}{2^k} : k\geq 1 \right\}.$$ We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$.
Anthony Gauvan

17 December 2023 hal04349073
We prove that the set of singular times for weak solutions of the homogeneous Boltzmann equation with very soft potentials constructed as in Villani (1998) has Hausdorff dimension at most $\frac{\gamma+2s}{2s}$ with $\gamma \in [4s,2s)$ and $s \in (0,1)$.
François Golse, Cyril Imbert, Luis Silvestre

12 December 2023 hal04119856
We present a rigorous mathematical analysis of the modeling of inviscid water waves. The freesurface is described as a parametrized curve. We introduce a numerically stable algorithm which accounts for its evolution with time. The method is shown to converge using approximate solutions, such as Stokes waves and GreenNaghdi solitary waves. It is finally tested on a wave breaking problem, for which an oddeven coupling suffices to achieve numerical convergence up to the splash without the need for additional filtering.
Emmanuel Dormy, Christophe Lacave

6 December 2023 hal04326021
Effects of atmospheric stratification and jet position on the properties of early aircraft contrailsThe net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the jets causing these contrails and the aircraft wake leads to large modifications in the contrail's altitude (ranging in hundreds of meters). This change in altitude heavily influences the net impact of contrails to global climate change, since the related change in temperature affects the ice content and the radiative properties of these contrails. This vortex entrainment strongly depends on the relative positioning of the jet with respect to the tip vortices. Furthermore, after the formation of the tip vortices and the turbulent diffusion of the jet, the jet dispersion is also driven by buoyant forces associated with atmospheric stratification. Here we focus on the vortex entrainment and buoyancy effects by running a large number of twodimensional simulations, scaled by the BruntVäisälä frequency and jet to tip vortex spacing, of the flow from the aftermath of the jet turbulent diffusion, vortex rollup and initial ice formation up to the vortex destabilization stage. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation. Eventually the results also explain several real life observations.
Pierre Saulgeot, Vincent Brion, Nicolas Bonne, Emmanuel Dormy, Laurent Jacquin

4 December 2023 hal03919233
Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales. In hydrodynamic turbulence, when the Reynolds number is large, the velocity field of the fluid becomes irregular and the rate of energy dissipation remains bounded from below even if the fluid viscosity tends to zero. A mathematical description of the turbulent cascade is a very active research topic since the pioneering work of Kolmogorov in hydrodynamic turbulence and that of Zakharov in wave turbulence. In both cases, these turbulent cascade mechanisms imply powerlaw behaviors of several statistical quantities such as power spectral densities. For a long time, these cascades were believed to be associated with nonlinear interactions, but recent works have shown that they can also take place in a dynamics governed by a linear equation with a differential operator of degree 0. In this spirit, we construct a linear equation that mimics the phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen as a linear transport equation, which corresponds to an operator of degree 0 in physical space. Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time.
Gabriel B. Apolinário, Geoffrey Beck, Laurent Chevillard, Isabelle Gallagher, Ricardo Grande

30 November 2023 hal04315916
We provide non asymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax optimal rate for estimating a certain probability measure $\mu$ with support in $\mathbb{R}^p$. The probability $\mu$ is considered to be the push forward of the Lebesgue measure on the $d$dimensional torus $\mathbb{T}^d$ by a map $g^\star:\mathbb{T}^d\rightarrow \mathbb{R}^p$ of smoothness $\beta+1$. Measuring the error with the $\gamma$Hölder Integral Probability Metric (IPM), we obtain up to logarithmic factors, the minimax optimal rate $O(n^{\frac{\beta+\gamma}{2\beta +d}}\vee n^{\frac{1}{2}})$ where $n$ is the sample size, $\beta$ determines the smoothness of the target measure $\mu$, $\gamma$ is the smoothness of the IPM ($\gamma=1$ is the Wasserstein case) and $d\leq p$ is the intrinsic dimension of $\mu$. In the process, we derive a sharp interpolation inequality between Hölder IPMs. This novel result of theory of functions spaces generalizes classical interpolation inequalities to the case where the measures involved have densities on different manifolds.
Arthur Stéphanovitch, Eddie Aamari, Clément Levrard

28 November 2023 hal04313048
We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $\mu$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $\mu$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,\mu)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and LyonsPeres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $\mu$ with a finite first moment.
Joshua Frisch, Eduardo Silva

28 November 2023 hal04313035
We study asymptotic invariants of metric spaces, defined in terms of the travelling salesman problem, and our goal is to classify groups and spaces depending on how well they can be ordered in this context. We characterize virtually free groups as those admitting an order which has some efficiency on $4$point subsets. We show that all $\delta$hyperbolic spaces can be ordered extremely efficiently, for the question when the number of points of a subset tends to $\infty$.
Anna Erschler, Ivan Mitrofanov

28 November 2023 hal04313029
We prove that finite entropy random walks on the torsionfree Baumslag group in dimension $d=2$ have nontrivial Poisson boundary. This is in contrast with the torsion case where the situation for simple random walks on Baumslag groups is the same as for the lamplighter groups of the same dimension. Our proof uses the realization of the Baumslag group as a linear group. We define and study a class of linear groups associated with multivariable polynomials which we denote $G_k(p)$. We show that the groups $G_3(p)$ have nontrivial Poisson boundary for all irreducible finite entropy measures, under a condition on the polynomial $p$ which we call the spaced polynomial property. We show that the Baumslag group has $G_3(1+xy)$ as a subgroup, and that the polynomial $p = 1+xy$, satisfies this property. Given any uppertriangle group of characteristic zero, we prove that one of the following must hold: 1) all finite second moment symmetric random walks on $G$ have trivial boundary 2) the group admits a block, which has a $3$ dimensional wreath product as a subgroup, and all nondegenerate random walks on $G$ have nontrivial boundary. 3) $G$ has a group $G_3(p)$ as a subgroup. We give a conjectural characterisation of all polynomials satisfying the spaced polynomial property. If this is confirmed, our result provides a characterisation of linear groups $G$ which admit a finitely supported symmetric random walk with nontrivial boundary.
Anna Erschler, Josh Frisch, Mark Rychnovsky

27 November 2023 hal04309737
We prove an elimination of imaginaires results for (almost all) henselian valued fields of equicharacteristic zero. To do so, we consider a mix of sorts introduced in earlier works of the two authors and define a generalized version of the klinear sorts. Under technical assumptions on the value group we prove that the imaginaries of such a valued field can be elimininated in the field, the klinear sorts and the imaginaries of the value group.
Silvain RideauKikuchi, Mariana Vicaría

27 November 2023 hal04309701
These are the notes (in French) for a Bourbaki talk on the twisted LangWeil estimates of Hrushovski, following a recent proof of Shuddhodan and Varshavsky.
Silvain RideauKikuchi

27 November 2023 hal04309633
Pseudo algebraically closed, pseudo real closed, and pseudo $p$adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for studying them: the class of pseudo $T$closed fields, where $T$ is an enriched theory of fields. These fields verify a "localglobal" principle for the existence of points on varieties with respect to models of $T$. This approach also enables a good description of some fields equipped with multiple $V$topologies, particularly pseudo algebraically closed fields with a finite number of valuations. One important result is a (model theoretic) classification result for bounded pseudo $T$closed fields, in particular we show that under specific hypotheses on $T$, these fields are NTP$_2$ of finite burden.
Samaria Montenegro, Silvain RideauKikuchi

27 November 2023 hal04308215
Anosov representations $\rho$ of a hyperbolic group $\Gamma$ into a semisimple Lie group $G$ are known to admit cocompact domains of discontinuity in flag varieties $G/Q$, endowing the compact quotient manifolds $M_\rho$ with a $(G,G/Q)$structure. In general the topology of $M_\rho$ can be quite complicated. In this article, we consider the case when $\Gamma$ is the fundamental group of a closed (real or complex) hyperbolic manifold $N$ and $\rho$ is a deformation of a (twisted) lattice embedding $\Gamma \to \mathrm{Isom}(\mathbf H_{\mathbb K})\to G$ through Anosov representations. We prove that, in this situation, $M_\rho$ is alway a smooth fiber bundle over $N$. Determining the topology of the fiber seems hard in general. The second part of the paper focuses on the special case when $N$ is a surface, $\rho$ a quasiHitchin representation into $\mathrm{Sp}(4,\mathbb C)$, and $M_\rho$ is modeled on the space of complex Lagrangians in $\mathb C^4$. We show that, in this case, the fiber is homeomorphic to $\mathbb C \mathbf P^2 \sharp \overline{\mathbb C \mathb P}^2$.
Daniele Alessandrini, Sara Maloni, Nicolas Tholozan, Anna Wienhard

27 November 2023 hal04308214
We introduce and study simple Anosov representations of closed hyperbolic surface groups, analogous to Minsky's primitive stable representations of free groups. We prove that the set of simple Anosov representations into $\mathrm{SL}(d, \mathbb C)$ with $d\geq 4$ strictly contains the set of Anosov representations. As a consequence, we construct domains of discontinuity for the mapping class group action on character varieties which contain nondiscrete representations.
Nicolas Tholozan, Tianqi Wang

24 November 2023 hal04306002
We consider hyperbolic and antide Sitter (AdS) structures on $M\times (0,1)$, where $M$ is a $d$dimensional GromovThurston manifold. If $M$ has cone angles greater than $2\pi$, we show that there exists a "quasifuchsian" (globally hyperbolic maximal) AdS manifold such that the future boundary of the convex core is isometric to $M$. When $M$ has cone angles less than $2\pi$, there exists a hyperbolic end with boundary a concave pleated surface isometric to $M$. Moreover, in both cases, if $M$ is a GromovThurston manifold with $2k$ pieces (as defined below), the moduli space of quasifuchsian AdS structures (resp. hyperbolic ends) satisfying this condition contains a submanifold of dimension $2k3$. When $d=3$, the moduli space of quasifuchsian AdS (resp. hyperbolic) manifolds diffeomorphic to $M\times (0,1)$ contains a submanifold of dimension $2k2$, and extends up to a "Fuchsian" manifold, that is, an AdS (resp. hyperbolic) warped product of a closed hyperbolic manifold by~$\R$. We use this construction of quasifuchsian AdS manifolds to obtain new compact quotients of $\O(2d,2)/\U(d,1)$. The construction uses an explicit correspondence between quasifuchsian $2d+1$dimensional AdS manifolds and compact quotients of $\O(2d,2)/\U(d,1)$ which we interpret as the space of timelike geodesic Killing fields of $\AdS^{2d+1}$.
Daniel Monclair, JeanMarc Schlenker, Nicolas Tholozan

16 November 2023 hal04289289
The ability to align points across two related yet incomparable point clouds (e.g. living in different spaces) plays an important role in machine learning. The GromovWasserstein (GW) framework provides an increasingly popular answer to such problems, by seeking a lowdistortion, geometrypreserving assignment between these points. As a nonconvex, quadratic generalization of optimal transport (OT), GW is NPhard. While practitioners often resort to solving GW approximately as a nested sequence of entropyregularized OT problems, the cubic complexity (in the number $n$ of samples) of that approach is a roadblock. We show in this work how a recent variant of the OT problem that restricts the set of admissible couplings to those having a lowrank factorization is remarkably well suited to the resolution of GW: when applied to GW, we show that this approach is not only able to compute a stationary point of the GW problem in time $O(n^2)$, but also uniquely positioned to benefit from the knowledge that the initial cost matrices are lowrank, to yield a linear time $O(n)$ GW approximation. Our approach yields similar results, yet orders of magnitude faster computation than the SoTA entropic GW approaches, on both simulated and real data.
Meyer Scetbon, Gabriel Peyré, Marco Cuturi

16 November 2023 hal04289279
Neural Ordinary Differential Equations (Neural ODEs) are the continuous analog of Residual Neural Networks (ResNets). We investigate whether the discrete dynamics defined by a ResNet are close to the continuous one of a Neural ODE. We first quantify the distance between the ResNet's hidden state trajectory and the solution of its corresponding Neural ODE. Our bound is tight and, on the negative side, does not go to 0 with depth N if the residual functions are not smooth with depth. On the positive side, we show that this smoothness is preserved by gradient descent for a ResNet with linear residual functions and small enough initial loss. It ensures an implicit regularization towards a limit Neural ODE at rate 1 over N, uniformly with depth and optimization time. As a byproduct of our analysis, we consider the use of a memoryfree discrete adjoint method to train a ResNet by recovering the activations on the fly through a backward pass of the network, and show that this method theoretically succeeds at large depth if the residual functions are Lipschitz with the input. We then show that Heun's method, a second order ODE integration scheme, allows for better gradient estimation with the adjoint method when the residual functions are smooth with depth. We experimentally validate that our adjoint method succeeds at large depth, and that Heun method needs fewer layers to succeed. We finally use the adjoint method successfully for finetuning very deep ResNets without memory consumption in the residual layers.
Michael E. Sander, Pierre Ablin, Gabriel Peyré

16 November 2023 hal04289271
Nonsmooth optimization is a core ingredient of many imaging or machine learning pipelines. Nonsmoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, lowrank and sharp edges. It is also the basis for the definition of robust loss functions and scalefree functionals such as squareroot Lasso. Standard approaches to deal with nonsmoothness leverage either proximal splitting or coordinate descent. These approaches are effective but usually require parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a different route, which operates a nonconvex but smooth overparameterization of the underlying nonsmooth optimization problems. This generalizes quadratic variational forms that are at the heart of the popular Iterative Reweighted Least Squares. Our main theoretical contribution connects gradient descent on this reformulation to a mirror descent flow with a varying Hessian metric. This analysis is crucial to derive convergence bounds that are dimensionfree. This explains the efficiency of the method when using small grid sizes in imaging. Our main algorithmic contribution is to apply the Variable Projection method which defines a new formulation by explicitly minimizing over part of the variables. This leads to a better conditioning of the minimized functional and improves the convergence of simple but very efficient gradientbased methods, for instance quasiNewton solvers. We exemplify the use of this new solver for the resolution of regularized regression problems for inverse problems and supervised learning, including total variation prior and nonconvex regularizers.
Clarice Poon, Gabriel Peyré