2025​​Activity reportProject-TeamPOPOPOP​​​‌

3.1 Repulsiveness and rigidity​​

Our challenges mostly involve​​​‌ point processes that are​ in an intermediate state​‌ between fully random, as​​ Poisson point processes, and​​ quasideterministic such as perturbed​​​‌ lattices. Intuitively, they exhibit‌ some rigidity properties that‌​‌ we need to define​​ and explore mathematically. A​​​‌ fruitful concept in this‌ context is hyperuniformity. Hyperuniformity,‌​‌ viewed as a rigidity​​ property of random configurations,​​​‌ manifests when the fluctuations‌ of particles are orders‌​‌ of magnitude smaller than​​ those expected in a​​​‌ pure random system of‌ points. More formally, it‌​‌ means that the variance​​ of the number of​​​‌ points within a domain‌ $\Delta$ becomes negligible relative‌​‌ to the volume of​​ $\Delta$ as $\Delta$ becomes​​​‌ large. Historically, hyperuniformity has‌ been explored in the‌​‌ context of studying the​​ compressibility of matter 40​​​‌, 46. Nevertheless,‌ understanding the order of‌​‌ the variance of the​​ number of points holds​​​‌ interdisciplinary significance. In material‌ science, hyperuniformity plays a‌​‌ crucial role in characterizing​​ naturally organized structures, such​​​‌ as crystals and quasicrystals‌ 47; in statistical‌​‌ physics, it appears necessary​​ to achieve equilibrium in​​​‌ charged fluids due to‌ the strong repulsion of‌​‌ the Coulomb interaction 43​​, 45; in​​​‌ data science, low fluctuations‌ of random inputs is‌​‌ known to improve the​​ convergence rate of Monte​​​‌ Carlo algorithms 28.‌ In all these settings,‌​‌ it seems that hyperuniformity​​ emerges primarily when the​​​‌ random points of the‌ configuration strongly repel each‌​‌ other, while being compelled​​ to remain close to​​​‌ each other by confining‌ forces. To thoroughly understand‌​‌ the relationship between repulsiveness​​ and rigidity is our​​​‌ primary objective in this‌ axis.

Another key concept‌​‌ in this context is​​ number-rigidity 38, 39​​​‌: a point process‌ is said to be‌​‌ number-rigid if, for any​​ bounded set $\Delta$,​​​‌ the number of points‌ within $\Delta$ is almost‌​‌ surely determined by the​​ knowledge of the point​​​‌ configuration outside $\Delta$.‌ At first glance, it‌​‌ may seem that only​​ the perturbed lattices, resembling​​​‌ a crystal-like structure, may‌ inherit this property. However,‌​‌ this is not the​​ case, and a large​​​‌ class of interacting points‌ is known to satisfy‌​‌ this property, particularly point​​ processes with a strong​​​‌ repulsiveness property as in‌ the hyperuniform case. Understanding‌​‌ this number rigidity property​​ and its links with​​​‌ hyperuniformity is also an‌ important goal in this‌​‌ axis.

3.2 Percolation and​​ infinite directed geodesics

One​​​‌ may also consider random‌ graphs or other random‌​‌ geometric structures built (through​​ various geometrical rules) on​​​‌ the point processes under‌ consideration. Henceforth, a very‌​‌ natural question is percolation,​​ that is the emergence​​​‌ of an unbounded cluster‌ or the occurrence of‌​‌ infinite directed geodesics. Percolation​​ constitutes a vast and​​​‌ dynamic field of research,‌ centered around the exploration‌​‌ of connectivity properties within​​ random graphs. Its significance​​​‌ spans across diverse disciplines‌ including physics, biology, and‌​‌ computer science. In physics,​​ it serves as a​​​‌ fundamental tool for characterizing‌ phenomena such as fluid‌​‌ flow through porous materials​​ or the behavior of​​​‌ disordered systems. In computer‌ science, it encompasses the‌​‌ resilience of communication networks​​ and the dissemination of​​​‌ information within social networks.‌ In mathematics, the study‌​‌ of discrete, lattice-based models​​​‌ has been extremely successful​ and recognized by major​‌ international prizes. Continuum percolation​​ which gathers models based​​​‌ on point processes in​ a continuous setting, has​‌ been much less explored.​​ Most of the known​​​‌ results hold for an​ underlying Poisson point process.​‌ Leveraging our expertise in​​ point processes, we aim​​​‌ to investigate more realistic​ models, with interactions. Recently,​‌ such dependent continuum percolation​​ models have appeared as​​​‌ natural paradigms for Device-to-Device​ (D2D) networks; see 42​‌ and the recent work​​ of our team 3​​​‌. A very exciting​ perspective in this field​‌ consists in introducing dynamics​​ in the random network​​​‌ with moving users. This​ aspect is currently underrepresented​‌ in the existing literature​​ and its investigation will​​​‌ necessarily require new concepts​ since classical tools in​‌ percolation theory are purely​​ static. With a similar​​​‌ flavour, we are also​ interested in Boolean percolation​‌ with an underlying repulsive​​ point process such as​​​‌ the Ginibre ensemble.

In​ some cases, as the​‌ transmission of an information​​ towards a central node​​​‌ or the spreading of​ a rumor in a​‌ social network, it is​​ relevant to work with​​​‌ directed graphs: each edge​ $\{x,\mathrm{y‌}\}$ is now considered​​ with a direction, say​​​‌ from $x$ to $\mathrm{y}$, becoming $(\mathrm{x‌},y)$ and​​ meaning for instance that​​​‌ a tweet of $\mathrm{x}$ has been retweeted by​‌ $y$. The Directed​​ Spanning Forest (DSF) 27​​​‌ represents a complex system​ with strong geometric dependencies​‌ in which information travels​​ towards a targeted node.​​​‌ In this context, any​ statistics about infinite directed​‌ geodesics (as their number,​​ asymptotic directions, density or​​​‌ possible scaling limit) starting​ at a given node​‌ are very meaningful. Several​​ results of this kind​​​‌ have been already obtained​ by members of the​‌ team for the DSF​​ 6, 33 and​​​‌ its hyperbolic version 4​, 14. Since​‌ there is no reason​​ for a rumor to​​​‌ spread according to a​ specific direction, we also​‌ investigate isotropic exploration of​​ node sets. A way​​​‌ to do it is​ to connect each node​‌ $x$ to its $\mathrm{\ell }\left(x\right)$-th​​​‌ closest node (according to​ random labels $\ell (‌·)$). Percolation​​ properties of the resulting​​​‌ graph have been investigated​ by members of the​‌ team in an Euclidean​​ setting. To mimick the​​​‌ structure of real-world social​ networks, a promising perspective​‌ would be to study​​ an hyperbolic version of​​​‌ this model.

3.3 Concentration​ and sampling

In many​‌ areas of applied science​​ and engineering, for example​​​‌ in Bayesian statistics, one​ is led to compute​‌ integrals of the form​​ $\int fd\mathrm{\mu ‌}$ of a given function​ $f$ with respect to​‌ a specified measure $\mathrm{\mu }$. It is of​​​‌ particular interest in high​ dimensions, and difficult even​‌ for simple measures $\mathrm{\mu }$ such as uniform measures​​​‌ on a convex body.​ Monte-Carlo algorithms for integration​‌ aims at giving numerical​​ approximations of the integral​​​‌ $\int fd\mathrm{\mu }$ by sums of the​‌ form ${\sum }_{i=1}^{n}f(x_i)$ (or​​​‌ weighted versions of the‌ latter), where $({\mathrm{x‌‌}}_1,...,x_n)$ is​​​‌ a well-chosen (random) sample‌ of points. In standard‌​‌ methods, the points are​​ usually chosen to be​​​‌ independent and identically distributed‌ (i.i.d.) random variables with‌​‌ distribution $\mu$. In​​ this case, the behavior​​​‌ of the empirical measure‌ distribution of the ${\mathrm{x‌‌}}_i$'s, its fluctuations​​ and concentration properties are​​​‌ well understood through the‌ classical law of large‌​‌ numbers, central limit theorem​​ and the machinery of​​​‌ concentration inequalities.

It is‌ a very natural idea‌​‌ to replace the i.i.d.​​ sample $(x_{\mathrm{1‌}},...,{\mathrm{x‌}}_n)$ by other‌​‌ point processes: introducing repulsiveness​​ between points should improve​​​‌ the exploration of space‌ and lead to open‌​‌ the path to better​​ concentration properties towards $\mathrm{\mu ‌}.$ As a consequence,‌ this should reduce the‌​‌ sample size required to​​ achieve a desired precision​​​‌ in the approximation of‌ the integral. However, this‌​‌ efficiency gain may come​​ at the cost of​​​‌ increased sampling complexity, as‌ point processes with interactions‌​‌ pose greater challenges in​​ sampling compared to i.i.d.​​​‌ samples. It raises questions‌ for which very few‌​‌ answers are known. For​​ a given measure $\mathrm{\mu ‌},$ how to generically‌ design point processes for‌​‌ which ${\sum }_{i=1}^{n}f(‌x_i)$ converges‌ at high speed towards‌​‌ the integral ? What​​ is the link between​​​‌ the law of the‌ point processes and the‌​‌ concentration properties of the​​ empirical measure around the​​​‌ target measure $\mu$ ?‌ How difficult is it‌​‌ to efficiently sample such​​ point processes ?

A​​​‌ few results in this‌ direction are already available‌​‌ in the literature. Our​​ team has been involved​​​‌ in showing concentration inequalities‌ for Coulomb gases with‌​‌ confining potential 2.​​ However, the links between​​​‌ the choice of the‌ potential and the limiting‌​‌ target measure is not​​ fully understood yet. Other​​​‌ natural candidates would be‌ log-gases, determinantal processes, Ries‌​‌ gases, and there are​​ a lot of mathematical​​​‌ questions to explore around‌ the concentration properties of‌​‌ those. Moreover, even if​​ we can design point​​​‌ processes well-adapted to a‌ given integration task, it‌​‌ can be challenging to​​ sample them efficiently. Numerical​​​‌ studies for Coulomb and‌ log-gases, as well as‌​‌ propositions of new simulation​​ algorithms based on Langevin​​​‌ dynamics, appear in 30‌. The quality of‌​‌ Langevin-based simulation methods were​​ investigated in 29,​​​‌ 36 but under hypotheses‌ that are generally not‌​‌ satisfied by Coulomb or​​ log-gases. Our team is​​​‌ involved in exploring such‌ questions in collaboration with‌​‌ Rémi Bardenet (CNRS, CRIStAL,​​ Lille University), in particular​​​‌ through a co-advised PhD‌ thesis funded by Rémi‌​‌ Bardenet's ERC starting grant​​ project Blackjack.

4.1 Metamaterials

Participants:‌ David Dereudre.

In‌​‌ the realm of materials​​ engineering, a significant challenge​​​‌ lies in crafting materials‌ with the capability to‌​‌ effectively absorb electromagnetic, acoustic,​​ or optical waves across​​​‌ a wide, predefined bandwidth.‌ The success in achieving‌​‌ these macroscopic properties is​​​‌ intricately tied to the​ microscopic arrangements of resonators​‌ embedded within the material​​ 35. This pursuit​​​‌ not only holds fundamental​ importance but also underscores​‌ its potential applications in​​ diverse fields, such as​​​‌ telecommunications, sensing technologies, and​ energy harvesting. Our contribution​‌ in this field is​​ in collaboration with teams​​​‌ at IEMN (Institute of​ Electronics, Microelectronics and Nanotechnology,​‌ Lille University).

4.2 D2D​​ networks

Participants: David Coupier​​​‌, Benoît Henry.​

In the future, networks​‌ and their infrastructures must​​ address numerous constraints arising​​​‌ from the demand for​ low-latency networks to support​‌ emerging applications, cope with​​ the escalating volume of​​​‌ data, all the while​ minimizing the environmental impact​‌ of these networks. In​​ this context, it is​​​‌ likely that the modeling​ of Device-to-Device (D2D) networks,​‌ including mobile agents, and​​ the analysis of such​​​‌ models will prove valuable​ in the efficient design​‌ and operation of networks.​​ This work has already​​​‌ started in the context​ of the Beyond 5G​‌ project in which IMT​​ is involved: see 3​​​‌. It will benefit​ from the IMT environment​‌ with many experts covering​​ all non-mathematical aspects of​​​‌ telecommunication networks and historic​ collaborations with the French​‌ telecommunications companies. Understanding how​​ long-range connections in a​​​‌ D2D network evolve when​ users are now allowed​‌ to move in the​​ urban media is a​​​‌ very natural, interesting and​ challenging question. Starting from​‌ the static picture of​​ a percolating network at​​​‌ time $t=\mathrm{0}$ and letting users move​‌ when time goes, one​​ may wonder how does​​​‌ the moving infinite clusters​ of users behave? Or,​‌ focusing on a mobile​​ and targeted user, is​​​‌ it often connected to​ the moving infinite cluster​‌ (which ensures a long-range​​ connection)? Let us finally​​​‌ point out that the​ possibility of long-range connection​‌ via D2D networks is​​ of capital importance for​​​‌ these companies. Indeed, such​ a network relying on​‌ the users themselves, would​​ only require (for the​​​‌ company) a small investment​ in the network infrastructure.​‌ This possibility of having​​ a functional network without​​​‌ prior investment constitutes an​ opportunity for new companies​‌ or operators to enter​​ the market and therefore​​​‌ a real economic threat​ for historic companies.

4.3​‌ Public health and Sociology​​

Participants: Viet Chi Tran​​​‌.

We have ongoing​ projects with sociologists and​‌ doctors (especially from Inserm​​ and Lille CHU) on​​​‌ the role of social​ networks in the propagation​‌ of epidemics, with a​​ particular interest in networks​​​‌ of people who inject​ drugs (PWID), opioids users​‌ and sexual networks. Addiction,​​ overdoses and infectious diseases​​​‌ that play a major​ role in the mortality​‌ and morbidity burden in​​ France and abroad are​​​‌ major societal topics. Nevertheless,​ the methodologies developed can​‌ be generalized to other​​ epidemics or applications. Social​​​‌ networks contribute to the​ dynamics of epidemics in​‌ different ways, either because​​ the disease itself spreads​​​‌ across the population by​ means of contacts between​‌ individuals (e.g. 31,​​ 7) or because​​​‌ the information shared along​ social networks can modify​‌ the population behaviours: we​​ consider to include rumors​​ and fake news on​​​‌ the one hand; prevention‌ and information about at-risk‌​‌ practices and contexts on​​ the other hand. Using​​​‌ mathematical models, our interest‌ is to understand how‌​‌ the geometry, the topology​​ of the graph and​​​‌ the interactions between individuals‌ shape the dynamic of‌​‌ the subgraph of people​​ contaminated or reached by​​​‌ the information or the‌ disease, and how the‌​‌ propagation of the latter​​ process can in turn​​​‌ modify the structure of‌ the underlying network. Our‌​‌ past collaborations already lead​​ to the launch of​​​‌ a `respondent sampling' survey‌ in 2018, a technique‌​‌ first used in France​​ to our knowledge, with​​​‌ the purpose of inferring‌ the social network of‌​‌ people who inject drugs.​​ Two PhD theses and​​​‌ several papers followed from‌ this collaboration (e.g. 34‌​‌, 32). With​​ the Covid 19 crisis,​​​‌ several scientific groups, in‌ France or abroad, have‌​‌ worked on modelling the​​ spread of diseases on​​​‌ networks. Most of the‌ points of view developed‌​‌ are on (stochastic) dynamical​​ systems ($R_{\mathrm{0‌}}$ considerations), statistical analysis or‌ numerical simulations. The subjects‌​‌ of PoPoPoP bring new​​ and interesting ways to​​​‌ handle these questions, and‌ we benefit from the‌​‌ strong interactions with sociologists.​​

A workgroup on the​​​‌ interface between mathematics and‌ biology and medicine has‌​‌ been created in September​​ by C. Tran with​​​‌ Olivier Bou-Aziz and François‌ Bachoc (University of Lille).‌​‌

4.4 Ecology and evolution​​

Participants: Benoît Henry,​​​‌ Mylène Maïda, Viet‌ Chi Tran.

We‌​‌ are also involved with​​ biologists (mainly of institutions​​​‌ participating to the Chaire‌ MMB, including members of‌​‌ the EEP laboratory at​​ Université of Lille, and​​​‌ of the IEES lab.‌ at Paris Sorbonne Université)‌​‌ on modelling the evolutions​​ of populations structured by​​​‌ a trait that affects‌ their reproductive and survival‌​‌ abilities. We have two​​ main applications in view:​​​‌ 1) the description of‌ pathogens and viruses evolution‌​‌ with in particular the​​ development of antibiotic resistances,​​​‌ 2) the conservation of‌ biodiversity. In both cases,‌​‌ our understanding can be​​ improved by the observation​​​‌ and study of the‌ past history and phylogenies‌​‌ (see e.g. 1,​​ 9). An important​​​‌ ingredient to take into‌ account are the interactions‌​‌ (selection, mutualism, predation etc.)​​ between the individuals. The​​​‌ latter are shaped by‌ the traits in the‌​‌ population, that can vary​​ from an individual to​​​‌ the other, and influence‌ in return the traits‌​‌ evolution. From individual-centered random​​ model where the dynamics​​​‌ is described at the‌ level of individuals (birth,‌​‌ death, horizontal transfer, mutation)​​ by mean of stochastic​​​‌ differential equations driven by‌ point processes, we can‌​‌ describe the evolution and​​ the extinction or conservation​​​‌ of the species in‌ large ecological network at‌​‌ the macroscopic population level​​ (see the review in​​​‌ 26). In the‌ framework of the Chaire‌​‌ MMB, we are often​​ lead to discuss with​​​‌ biologists and companies interested‌ in these questions (Veolia,‌​‌ EDF and others).

4.5​​ Monte-Carlo integration and sampling​​​‌

Participants: Mylène Maïda,‌ Martin Rouault.

Putting‌​‌ everything together in the​​​‌ Concentration and sampling axis​ of the research program​‌ would lead long term​​ to be able, for​​​‌ a large number of​ numerical integration tasks, to​‌ design repulsive processes that​​ are well adapted to​​​‌ the task and that​ we know how to​‌ sample efficiently. In addition,​​ we aim at providing​​​‌ a library implementing these​ procedures, in open access​‌ and intended to mathematicians​​ and physicists, as well​​​‌ as research engineers.

4.6​ Denoising of signals

Participants:​‌ Raphaël Butez.

Denoising​​ techniques based on the​​​‌ zeros of the spectrogram​ of the signal were​‌ introduced by Flandrin 37​​ and have been studied​​​‌ over the past 10​ years with great success.​‌ These new techniques are​​ mostly empirical and no​​​‌ theoretical studies guarantee they​ exist. As the spectrogram​‌ of the white noise​​ is essentially the flat​​​‌ Gaussian Analytic Function, a​ good understanding of its​‌ zeros will allow us​​ to refine the existing​​​‌ techniques and to provide​ criteria for a good​‌ reconstruction of the signal.​​ More precisely, we want​​​‌ to obtain bounds on​ the signal-to-noise ratio which​‌ ensure a good reconstruction​​ with controlled probability of​​​‌ failure.

5.1 IDLA in ${\mathrm{\mathbb{Z}‌}}^d$ with infinitely many​​ sources

Participant: David Coupier​​​‌.

According to an​ IDLA (Internal Diffusion Limited​‌ Aggregation) protocol, when a​​ source launches a particle,​​​‌ this one performs a​ simple random walk on​‌ ${\mathbb{Z}}^d$ until exiting​​ the current aggregate $\mathrm{A‌}$ through a site $\mathrm{x}$ which is added to​‌ the aggregate: $A$ is​​ updated to $A\cup ‌\left\{x\right\}$.​ In 12, we​‌ consider a random growth​​ model based on the​​​‌ IDLA protocol with infinitely​ many sources from a​‌ hyperplane of ${\mathbb{Z}}^{\mathrm{d}}$, say $\mathscr{H}$.​​​‌ Precisely, $n$ particles are​ launched from each source​‌ of $\mathscr{H}$ according to​​ a prescribed order. This​​​‌ process provides an infinite​ aggregate $A_n[‌\infty ]$ for which​​ we prove a shape​​​‌ theorem as $n\to \infty$, in any​‌ dimension $d\ge \mathrm{2}$, with sublogarithmic fluctuations.​​​‌

In 20, we​ use i.i.d. exponential clocks​‌ to launch the particles​​ during a time interval​​​‌ $[0,\mathrm{n}]$, so that​‌ the next source emitting​​ a particle is choosen​​​‌ uniformly within the (infinite)​ set $\mathscr{H}$. Also,​‌ we not only retain​​ the site $x$ by​​​‌ which a given particle​ exits the current aggregate​‌ but also the edge​​ used by the particle​​​‌ to reach $x$.​ This construction leads to​‌ a random forest, denoted​​ by ${ℱ}_n[‌\infty ]$ and called​ the IDLA forest. With​‌ respect to the aggregate​​ $A_n\left[\mathrm{\infty ‌}\right]$, the construction​ of the IDLA forest​‌ ${ℱ}_n\left[\mathrm{\infty }\right]$ is much more​​​‌ intricated. Hence, we state​ a stabilization result for​‌ ${ℱ}_n\left[\mathrm{\infty }\right]$, based on​​​‌ tools coming from the​ percolation theory, which allows​‌ us to define properly​​ the IDLA forest as​​​‌ an ergodic random graph.​

5.2 Percolation of the​‌ two-neighbor graph on the​​ planar lattice

Participants: David​​ Coupier, Benoît Henry​​​‌.

One of the‌ aims of the PoPoPoP‌​‌ team is to investigate​​ how dependencies can affect​​​‌ large scale connectivity in‌ random graph. A first‌​‌ fundamental question is whether​​ the imposition of rigid​​​‌ local constraints facilitates or‌ hinders percolation compared to‌​‌ standard models. A model​​ introduced recently concerns graphs​​​‌ with fixed out-degrees, known‌ as $k$-neighbor graphs,‌​‌ where each vertex connects​​ to exactly $k$ nearest​​​‌ neighbors chosen uniformly at‌ random. This creates a‌​‌ degenerate random environment with​​ specific local dependencies. In​​​‌ a collaboration with Benedikt‌ Jahnel and Jonas Köppl‌​‌ (WIAS), we investigated the​​ behavior of this model​​​‌ in the planar case.‌ In the preprint 5‌​‌, we settle a​​ conjecture regarding the case​​​‌ $k=2$ in‌ dimension $d=\mathrm{2‌‌}$. We prove that​​ the planar 2-neighbor graph​​​‌ percolates, meaning the origin‌ is connected to infinity‌​‌ with positive probability. Our​​ proof relies on a​​​‌ specific exploration algorithm and‌ a comparison to i.i.d.‌​‌ bond percolation, demonstrating mathematically​​ that the rigid constraint​​​‌ of having exactly 2‌ neighbors reduces the randomness‌​‌ in a way that​​ is strictly beneficial for​​​‌ percolation compared to average-degree‌ equivalents.

5.3 Line-of-sight percolation‌​‌ on Poisson-Delaunay triangulations

Participants:​​ David Coupier, Benoît​​​‌ Henry.

Stochastic geometry‌ is a key tool‌​‌ for analyzing the macroscopic​​ properties of telecommunication networks.​​​‌ In D2D networks, signal‌ propagation is heavily constrained‌​‌ by the street layout,​​ which dictates line-of-sight availability.​​​‌ In the work 3‌, in collaboration with‌​‌ D. Corlin Marchand (former​​ Postdoc at IMT Nord​​​‌ Europe), we proposed a‌ model where the street‌​‌ system is represented by​​ a Poisson-Delaunay triangulation. Users​​​‌ are distributed according to‌ a Cox process supported‌​‌ on the edges (streets)​​ and vertices (crossroads) of​​​‌ this triangulation. We study‌ the associated connectivity graph,‌​‌ where links are formed​​ based on distance and​​​‌ alignment along the streets.‌ We establish a complete‌​‌ phase diagram of the​​ model which improves the​​​‌ results existing on similar‌ existing models, eliminating the‌​‌ unknown intermediate regions often​​ found in previous works.​​​‌

5.4 Measure estimation on‌ a manifold explored by‌​‌ a diffusion process

Participant:​​ Viet Chi Tran.​​​‌

In the article 16‌, we consider the‌​‌ problem of estimating the​​ stationary measure $\mu$ of​​​‌ a diffusion process ${(‌X_t)}_{\mathrm{t‌‌}\in [0,T]}$ on a​​​‌ compact connected $d$-dimensional‌ manifold $ℳ$ without boundary,‌​‌ from the observation of​​ one of its path.​​​‌ We base our estimation‌ on the occupation measure‌​‌ ${\mu }_T$ of this​​ process defined for all​​​‌ bounded measurable test function‌ $f$ by

Wang and Zhu 48‌ showed that for the‌​‌ Wasserstein metric $W_{\mathrm{2}}$ and for $d\ge ‌5$, the convergence‌ rate of $T^{-‌‌1/(\mathrm{d}-2)}$ is​​​‌ attained when ${\left({\mathrm{X‌}}_t\right)}_{t\in ‌‌[0,\mathrm{T‌}]}$ is a Langevin​ diffusion. We extend their​‌ result in several directions.​​ First, we show that​​​‌ the rate of convergence​ holds for a large​‌ class of diffusion paths,​​ whose generators are uniformly​​​‌ elliptic. Second, the regularity​ of the density $\mathrm{p‌}$ of the stationary measure​​ $\mu$ with respect to​​​‌ the volume measure of​ $ℳ$ can be leveraged​‌ to obtain faster estimators:​​ when $p$ belongs to​​​‌ a Sobolev space of​ order $\ell \ge \mathrm{2‌}$, smoothing the occupation​​ measure by convolution with​​​‌ a kernel yields an​ estimator whose rate of​‌ convergence is of order​​ $T^{-(\mathrm{\ell ‌}+1)/(2\ell +‌d-2)}$. We further show​​​‌ that this rate is​ the minimax rate of​‌ estimation for this problem.​​

5.5 Euclidean Directed Spanning​​​‌ Forest and Radial Spanning​ Tree

Participant: Tom Garcia-Sanchez​‌.

The Euclidean Directed​​ Spanning Forest (DSF) is​​​‌ a geometric random graph​ whose vertex set is​‌ given by a homogeneous​​ Poisson Point Process $\mathrm{𝒩‌}$ on ${ℝ}^d$,​ $d\ge 2$,​‌ and whose (directed) edges​​ consist of all pairs​​​‌ $(x,\mathrm{y})\in {𝒩}^{\mathrm{2‌}}$ such that $y$ is​​ the closest point to​​​‌ $x$ in $𝒩$ for​ the ${\ell }^p$ distance,​‌ $p\in [\mathrm{1},\infty ]$,​​​‌ among points with a​ strictly larger $e_{\mathrm{d‌}}$ coordinate (the $d$-th​​ vector of the canonical​​​‌ basis). This geometric random​ graph introduced by Baccelli​‌ and Bordenave in 27​​ in the case $\mathrm{p‌}=d=\mathrm{2}$ admits a natural forest​‌ structure: it is a​​ collection of unrooted directed​​​‌ trees. In 22,​ we prove that for​‌ $p\in \{\mathrm{1},2,\mathrm{\infty ‌}\}$, the graph​ is almost surely a​‌ tree when $d=3$, and consists​​​‌ of infinitely many disjoint​ trees when $d\ge ‌4$. Additionally, we​​ show that for all​​​‌ $p\in [\mathrm{1},\infty ]$,​‌ the DSF in dimension​​ 2 is almost surely​​​‌ a tree and, under​ appropriate diffusive scaling, converges​‌ weakly to the Brownian​​ web, generalizing the result​​​‌ of 6 for $\mathrm{p}=2$.

The​‌ (Euclidean) Radial Spanning Tree​​ (RST) is a radial​​​‌ version of the DSF​ in which each vertex​‌ $x\in 𝒩$ has​​ a unique outgoing edge​​​‌ pointing to the nearest​ point in $𝒩\cup ‌\left\{0\right\}$ that​​ lies closer to the​​​‌ origin (w.r.t. the Euclidean​ distance ${\ell }^2$).​‌ By construction, it forms​​ almost surely a tree​​​‌ rooted at 0. In​ 23, we prove​‌ that the RST is​​ almost surely straight in​​​‌ any dimension $d\ge 2$, meaning roughly​‌ that its subtrees become​​ thinner and thinner as​​​‌ their roots move away​ from the origin. As​‌ a by-product, using the​​ strategy developed by Howard​​​‌ and Newman 41,​ we obtain that with​‌ probability 1 each infinite​​ branch of the RST​​​‌ admits an asymptotic direction​ and each asymptotic direction​‌ is targeted by (at​​ least) one infinite branch.​​

5.6 Thick trace at​​​‌ infinity for the Hyperbolic‌ Radial Spanning Tree

Participants:‌​‌ David Coupier, Viet​​ Chi Tran.

In​​​‌ 14, we study‌ the hyperbolic Radial Spanning‌​‌ Tree (RST) in any​​ dimension $d\ge \mathrm{2‌}$. The hyperbolic RST‌ had already been introduced‌​‌ and studied by the​​ same authors in 4​​​‌. In this new‌ paper, we are interested‌​‌ in its fine topological​​ properties, and in particular,​​​‌ its exceptional directions which‌ are defined as the‌​‌ directions $\xi \in {\mathrm{𝕊}}^{d-1}$ targeted​​​‌ by at least two‌ infnite branches of the‌​‌ RST. Exploiting the features​​ of the hyperbolic geometry,​​​‌ we state that any‌ infinite subtree $T$ of‌​‌ the RST almost surely​​ admits a thick trace​​​‌ at infinity, i.e. the‌ set of directions $\mathrm{\xi ‌‌}\in {𝕊}^{d-1}$ targeted by an​​​‌ infinite branch of $\mathrm{T‌}$ has a positive measure.‌​‌ In the case $\mathrm{d}=2$, this​​​‌ solves the N3G problem‌ (for No 3 Geodesics‌​‌) for the hyperbolic​​ RST: the hyperbolic RST​​​‌ in dimension 2 does‌ not contain 3 infinite‌​‌ branches with the same​​ (random) asymptotic direction with​​​‌ probability one.

5.7 Gibbs‌ point processes for numerical‌​‌ integration

Participants: Mylène Maïda​​, Martin Rouault.​​​‌

Markov chain Monte Carlo‌ algorithms (MCMC) are numerical‌​‌ integration algorithms that are​​ ubiquitous in high-dimensional statistical​​​‌ inference and Bayesian machine‌ learning. The crux is‌​‌ to sample a carefully-chosen​​ Markov chain in the​​​‌ domain of integration, and‌ average the evaluations of‌​‌ the integrand along that​​ chain. However, estimators resulting​​​‌ from MCMC algorithms have‌ a mean squared error‌​‌ that decreases as $\mathrm{1}/n$, where​​​‌ $n$ is the number‌ of time steps in‌​‌ the Markov chain sample.​​ When the integrand function​​​‌ is very expensive to‌ evaluate, it is relevant‌​‌ to design an estimator​​ with better guarantees than​​​‌ MCMC for the same‌ number of evaluations of‌​‌ the function (giving the​​ same guarantees with less​​​‌ points), even if it‌ requires a higher computational‌​‌ budget to sample.

In​​ the framework of the​​​‌ PhD thesis of Martin‌ Rouault , co-supervised by‌​‌ R. Bardenet and Mylène​​ Maïda , we exploited​​​‌ the intuition that dependence,‌ or repulsiveness, between points‌​‌ brings qualitative variance reduction,​​ an idea which is​​​‌ a key point of‌ the third axis of‌​‌ PoPoPoP scientific project. In​​ particular, we propose to​​​‌ draw $n$ points according‌ to a Gibbs distribution‌​‌ and study the convergence​​ properties of the associated​​​‌ estimator for numerical integration.‌ In a first work‌​‌ in 2024, we prove​​ a fast concentration inequality​​​‌ for such Gibbs distributions‌ with general bounded pairwise‌​‌ interaction. Given a fixed​​ precision level on the​​​‌ integration error, this means‌ that a smaller number‌​‌ of points $n$ is​​ required to get the​​​‌ same confidence regions as‌ MCMC. In the communication‌​‌ 18, we illustrate​​ how this strategy can​​​‌ be used in the‌ context of Bayesian inference.‌​‌ Approximately sampling points from​​ this Gibbs distribution however​​​‌ requires computing the so-called‌ interaction potential of the‌​‌ target measure. In the​​​‌ preprint 19, we​ develop a two-step-procedure taking​‌ into account an empirical​​ approximation of the potential.​​​‌ The model is mathematically​ more involved but through​‌ a quenched large deviation​​ principle, we can show​​​‌ that the guarantees previously​ obtained still hold, at​‌ least asymptotically, including this​​ approximation.

5.8 Hyperuniform versus​​​‌ Poisson Distributions in Random​ Metasurfaces at Infrared Wavelengths​‌

Participant: David Dereudre.​​

Metamaterials are artificial materials​​​‌ made of very small​ structures that can control​‌ how light or other​​ electromagnetic waves behave. Unlike​​​‌ traditional optical materials, their​ properties come from the​‌ shape and arrangement of​​ these tiny elements, called​​​‌ meta-atoms, rather than from​ their chemical composition. Most​‌ early metamaterials were arranged​​ in very regular, periodic​​​‌ patterns. However, researchers have​ recently become interested in​‌ disordered arrangements, where meta-atoms​​ are placed in a​​​‌ more random way. This​ approach offers more flexibility​‌ in design and can​​ simplify fabrication, especially for​​​‌ large-area optical devices. One​ basic type of random​‌ arrangement follows a Poisson​​ distribution, similar to how​​​‌ raindrops fall on the​ ground. In this case,​‌ the meta-atoms can overlap,​​ which may strongly affect​​​‌ their optical response because​ each meta-atom acts like​‌ a small resonator whose​​ behavior depends on its​​​‌ size and shape.

Another​ type of disordered structure,​‌ called hyperuniform, lies between​​ perfect order and complete​​​‌ randomness. These structures are​ found in nature and​‌ have attracted attention in​​ many areas of wave​​​‌ physics, from acoustics to​ optics. Hyperuniform designs can​‌ prevent overlap between meta-atoms​​ while still being disordered,​​​‌ and they have already​ led to improved performance​‌ in several optical and​​ electromagnetic devices.

In 44​​​‌, in collaboration with​ colleagues at IEMN, we​‌ compare metasurfaces based on​​ Poisson and hyperuniform distributions.​​​‌ They study how these​ two types of disorder​‌ influence light extinction and​​ absorption in the infrared​​​‌ range. By analyzing how​ individual meta-atoms absorb light​‌ and how they interact​​ with each other, the​​​‌ study shows that both​ disordered designs can achieve​‌ strong and broadband optical​​ performance for realistic material​​​‌ densities. In 11,​ we analyze the extinction​‌ and absorption properties of​​ randomly distributed Metal-Insulator-Metal metasurfaces​​​‌ operating at an infrared​ wavelength. Our attention has​‌ been focused on the​​ comparison between correlated disordered,​​​‌ namely, hyperuniform, and Poisson​ Point Process distributed structures.​‌ To this aim, we​​ define absorption and extinction​​​‌ terms by considering Floquet​ modes within a super-cell​‌ simulated by a finite​​ element approach. Then, the​​​‌ role of particle intercoupling​ in high-density random metasurfaces​‌ has been pointed out​​ and analyzed as a​​​‌ function of linear polarization​ of the impinging wave.​‌ Finally, hyperuniform and Poisson​​ Point Process distribution have​​​‌ been compared for metasurface​ filling fraction ranging between​‌ 5% and 50% and​​ relative standard deviations up​​​‌ to 1:0, taking the​ square periodic array as​‌ a reference.

5.9 (Non)-hyperuniformity​​ of perturbed lattices

Participant:​​​‌ David Dereudre.

In​ 8, we study​‌ hyperuniformity properties of random​​ point processes obtained by​​​‌ perturbing regular lattices in​ Euclidean space. Hyperuniformity is​‌ defined through the asymptotic​​ behavior of the number​​ variance in large observation​​​‌ windows: a point process‌ is said to be‌​‌ hyperuniform when density fluctuations​​ grow sublinearly with the​​​‌ volume. We start from‌ the fact that stationary‌​‌ lattices are extremal examples​​ of hyperuniform point configurations,​​​‌ exhibiting the slowest possible‌ growth of the number‌​‌ variance. We then investigate​​ whether this property is​​​‌ preserved when each lattice‌ point is randomly displaced,‌​‌ leading to a class​​ of perturbed lattices. The​​​‌ perturbations are assumed to‌ be identically distributed and‌​‌ stationary, while allowing for​​ arbitrary dependence. Our main​​​‌ results show that the‌ stability of hyperuniformity depends‌​‌ crucially on the dimension​​ and on the moment​​​‌ assumptions on the perturbations:‌ in dimensions one and‌​‌ two, if the perturbations​​ have a finite moment​​​‌ of order equal to‌ the dimension, the perturbed‌​‌ lattice remains hyperuniform. This​​ condition is essentially sharp,​​​‌ since weaker assumptions may‌ produce point processes that‌​‌ are not hyperuniform and​​ may even exhibit infinite​​​‌ number variance in bounded‌ regions. In dimensions three‌​‌ and higher, hyperuniformity is​​ much more fragile: we​​​‌ show that arbitrarily small‌ perturbations can already destroy‌​‌ hyperuniformity, leading to macroscopic​​ density fluctuations.

Concerning the​​​‌ stronger notion of class-I‌ hyperuniformity, we establish a‌​‌ sharp sufficient condition in​​ dimension one based solely​​​‌ on the size of‌ the perturbations, while we‌​‌ show that in dimension​​ two such conditions are​​​‌ no longer sufficient. Finally,‌ we prove that hyperuniformity‌​‌ does not impose a​​ universal decay rate for​​​‌ the rescaled number variance:‌ even within the class‌​‌ of hyperuniform perturbed lattices,​​ this decay can be​​​‌ arbitrarily slow. Overall, our‌ results provide a precise‌​‌ characterization of how random​​ perturbations affect large-scale regularity​​​‌ in point configurations, and‌ they highlight the delicate‌​‌ balance between disorder, dimension,​​ and correlation structure in​​​‌ hyperuniform systems.

5.10 Rigidity‌ of one-dimensional point processes‌​‌ via optimal transport

Participants:​​ Rafaël Digneaux, David​​​‌ Dereudre.

In 21‌, we investigate rigidity‌​‌ phenomena in one-dimensional point​​ processes. We show that​​​‌ the existence of an‌ $L^1$ transport map‌​‌ from a stationary lattice​​ or the Lebesgue measure​​​‌ to a point process‌ is sufficient to guarantee‌​‌ the properties of Number-Rigidity​​ and Cyclic-Factor. We then​​​‌ apply this result to‌ non-singular Riesz gases with‌​‌ parameter $s\in (-2,-‌1]$, defined‌ in infinite volume as‌​‌ accumulation points of stationarized​​ finite-volume Riesz gases. This​​​‌ includes, for $s=‌-1$, the‌​‌ well-known one-dimensional Coulomb gas​​ (also called Jellium plasma,​​​‌ or the one-component 1D‌ plasma).

5.11 Liquid-gas phase‌​‌ transition for Gibbs point​​ process with Quermass interaction​​​‌

Participant: David Dereudre.‌

In 15, we‌​‌ prove the existence of​​ a liquid-gas phase transition​​​‌ for continuous Gibbs point‌ process in ${ℝ}^{\mathrm{d‌‌}}$ with Quermass interaction. The​​ Hamiltonian we consider is​​​‌ a linear combination of‌ the volume $𝒱$,‌​‌ the surface measure $\mathrm{𝒮}$ and the Euler-Poincaré characteristic​​​‌ $\chi$ of a halo‌ of particles (i.e. an‌​‌ union of balls centred​​ at the positions of​​​‌ particles). We show the‌ non-uniqueness of infinite volume‌​‌ Gibbs measures for special​​​‌ values of activity and​ temperature, provided that the​‌ temperature is low enough.​​ Moreover we show the​​​‌ non-differentiability of the pressure​ at these critical points.​‌ Our main tool is​​ an adaptation of the​​​‌ Pirogov-Sinaï-Zahradnik theory for continuous​ systems with interaction exhibiting​‌ a saturation property.

5.12​​ Absence of percolation for​​​‌ infinite Poissonian systems of​ stopped paths

Participants: David​‌ Coupier, David Dereudre​​.

In 13,​​​‌ we study a stopped​ germ–grain model in the​‌ plane based on a​​ homogeneous Poisson point process,​​​‌ where each point is​ marked with an independent​‌ continuous path. From each​​ point, a grain grows​​​‌ along its path and​ stops at its first​‌ intersection with another grain.​​ Under a mild moment​​​‌ condition on the paths,​ this interacting growth process​‌ is well defined.

Our​​ main question concerns percolation​​​‌ of the random union​ of stopped grains. We​‌ introduce a geometric loop​​ condition, requiring that loops​​​‌ can form with positive​ probability at arbitrarily small​‌ space and time scales,​​ and we investigate whether​​​‌ this condition prevents percolation.​

We prove that, under​‌ the loop condition and​​ a finite second-moment assumption​​​‌ on the paths, the​ model does not percolate​‌ almost surely. Our result​​ substantially generalizes previous non-percolation​​​‌ results for thin-grain germ–grain​ models by allowing general​‌ continuous paths and by​​ weakening the integrability assumptions​​​‌ on the growth dynamics.​

5.13 Probabilistic methods for​‌ studying the partition function​​ of the two-dimensional Yang-Mills​​​‌ theory

Participant: Mylène Maïda​.

Getting asymptotic expansions​‌ for matrix integrals is​​ an active topic within​​​‌ random matrix theory. When​ the coefficients of the​‌ expansions are related to​​ geometrical or topological invariants,​​​‌ these expansions are called​ topological expansions. It is​‌ in general a hard​​ mathematical task to show​​​‌ that topological expansions are​ not only formal power​‌ series but that they​​ are properly convergent. In​​​‌ an ongoing collaboration with​ Thibaut Lemoine (Université de​‌ Strasbourg), we are interested​​ in a specific matrix​​​‌ integral, which happens to​ be the partition function​‌ of a model in​​ quantum field theory called​​​‌ the two-dimensional Yang-Mills theory.​ In 17, using​‌ probabilistic arguments on random​​ partitions, we get a​​​‌ full description of the​ topological expansion, when the​‌ underlying manifold is the​​ torus and the gauge​​​‌ group is the unitary​ group. In the recent​‌ preprint 25, we​​ generalize these results to​​​‌ other compact Lie groups​ and show that the​‌ corresponding asymptotic expansions are​​ related to the enumeration​​​‌ of ramified coverings of​ the torus. This leads​‌ to establishing rigorously a​​ string/gauge duality result predicted​​​‌ by physicists Gross and​ Taylor in the nineties.​‌

Although this line of​​ research is not directly​​​‌ related to the main​ lines of research of​‌ PoPoPoP, we want to​​ emphasize that the matrix​​​‌ integral giving the partition​ function of the studied​‌ model is closely related​​ to a diffusion over​​​‌ the groupe of unitary​ matrices, called the unitary​‌ Brownian motion, which exhibits​​ very interesting dependence and​​​‌ repulsiveness and has Gibbs​ measure of the type​‌ described in the previous​​ subsection as invariant measures.​​

5.14 Continuous limits of​​​‌ large plant-pollinator random networks‌ and some applications

Participant:‌​‌ Viet Chi Tran.​​

In 10, we​​​‌ study a stochastic individual-based‌ model of interacting plant‌​‌ and pollinator species through​​ a bipartite graph: each​​​‌ species is a node‌ of the graph, an‌​‌ edge representing interactions between​​ a pair of species.​​​‌ The dynamics of the‌ system depends on the‌​‌ between- and within-species interactions:​​ pollination by insects increases​​​‌ plant reproduction rate but‌ has a cost which‌​‌ can increase plant death​​ rate, depending on the​​​‌ densities of pollinators. Pollinators‌ reproduction is increased by‌​‌ the resources harvested on​​ plants. Each species is​​​‌ characterized by a trait‌ corresponding to its degree‌​‌ of generalism. This trait​​ determines the structure of​​​‌ the interaction graph and‌ the quantities of resources‌​‌ exchanged between species. Our​​ model includes in particular​​​‌ nested or modular networks.‌ Deterministic approximations of the‌​‌ stochastic measure-valued process by​​ systems of ordinary differential​​​‌ equations or integro-differential equations‌ are established and studied,‌​‌ when the population is​​ large or when the​​​‌ graph is dense and‌ can be replaced with‌​‌ a graphon. The long-time​​ behaviors of these limits​​​‌ are studied and central‌ limit theorems are established‌​‌ to quantify the difference​​ between the discrete stochastic​​​‌ individual-based model and the‌ deterministic approximations. Finally, studying‌​‌ the continuous limits of​​ the interaction network and​​​‌ the resulting PDEs, we‌ show that nested plant-pollinator‌​‌ communities are expected to​​ collapse towards a coexistence​​​‌ between a single pair‌ of species of plants‌​‌ and pollinators.

5.15 Goodness-of-fit​​ testing for the stationary​​​‌ density of a size-structured‌ PDE

Participant: Viet Chi‌​‌ Tran.

In 24​​, we consider two​​​‌ division models for structured‌ cell populations, where cells‌​‌ can grow, age and​​ divide. These models have​​​‌ been introduced in the‌ literature under the denomination‌​‌ of `mitosis' and `adder'​​ models. In the recent​​​‌ years, there has been‌ an increasing interest in‌​‌ biology to understand whether​​ the cells divide equally​​​‌ or not, as this‌ can be related to‌​‌ important mechanisms in cellular​​ aging or recovery. We​​​‌ are therefore interested in‌ testing the null hypothesis‌​‌ $H_0$ where the​​ division of a mother​​​‌ cell results into two‌ daughters of equal size,‌​‌ against the alternative hypothesis​​ $H_1$ where the​​​‌ division is asymmetric and‌ ruled by a kernel‌​‌ that is absolutely continuous​​ with respect to the​​​‌ Lebesgue measure. The sample‌ consists of i.i.d. observations‌​‌ of cell sizes and​​ ages drawn from the​​​‌ population, and the division‌ is not directly observed.‌​‌ The hypotheses of the​​ test are reformulated as​​​‌ hypotheses on the stationary‌ size and age distributions‌​‌ of the models, which​​ we assume are also​​​‌ the distributions of the‌ observations. We propose a‌​‌ goodness-of-fit test that we​​ study numerically on simulated​​​‌ data before applying it‌ on real data.

6.1 Bilateral​​​‌ contracts with industry

Participants:‌ Viet Chi Tran.‌​‌

• Viet Chi Tran works​​ with the enterprise Meteors​​​‌ (now part of Fourseeds‌) on the marketing‌​‌ impact on purchases of​​​‌ the irritation of customers​ when confronted to too​‌ massive mailings.

  Using a​​ stochastic individual-based model, we​​​‌ analyze the influence of​ marketing emails and SMS​‌ strategies on customer purchasing​​ behavior and decisions to​​​‌ unsubscribe. When used effectively,​ email and SMS campaigns​‌ can spark interest in​​ new products and encourage​​​‌ purchases. However, bombarding customers​ with excessive emails or​‌ messages can lead to​​ negative outcomes such as​​​‌ reduced brand engagement and​ environmental consequences. Using duration​‌ models and point processes,​​ we account for customers​​​‌ exposures to the brand​ which can differ from​‌ an individual to the​​ other and evolve in​​​‌ time. This exposure corresponds​ to an ad-stock point​‌ of view and is​​ influenced by the communication​​​‌ received, purchase activities, and​ decisions to unsubscribe. A​‌ customer's exposure level to​​ the brand affects their​​​‌ rate of purchases and​ unsubscription decisions. A working​‌ paper in collaboration with​​ the enterprise and Annabel​​​‌ Salerno (IAE Lille) is​ in progress. After discussing​‌ the model and its​​ statistical inference, we perform,​​​‌ as a case study,​ numerical simulations to explore​‌ the possible outcomes of​​ advertisement policies under various​​​‌ scenarios, including potential saturation​ or customer impatience and​‌ starting from actual data​​ from a leather goods​​​‌ company. In the past,​ a contract had been​‌ signed with Lille University.​​ A follow up with​​​‌ Inria is planned.

• In​ the context of the​‌ possible extension of the​​ Chaire MMB, in the​​​‌ steering committee of which​ Viet Chi Tran belongs,​‌ a workshop Modélisation mathématique​​ et biodiversité : enjeux​​​‌ et outils was organized​ with the purpose of​‌ reuniting colleagues from the​​ academy and enterprises. The​​​‌ talks alternated between presentation​ of private companies on​‌ their questions regarding biodiverity​​ and the associated modelling​​​‌ and of academics presenting​ possible solutions. Note that​‌ several Inria teams where​​ represented additionnally to PoPoPoP:​​​‌ Merge and Simba. The​ possible collaborations are still​‌ under discussions.

7.1 International​​​‌ initiatives

7.2 National initiatives

• Mylène​​ Maïda is a member​​​‌ of the ANR project​ Large Objects Under Combinatorial​‌ Constraints and Outside Uniform​​ Models (LOUCCOUM)​​​‌ , funded by the​ ANR (French National Research​‌ Agency)
• Viet Chi Tran​​ is in the steering​​​‌ committee of the Chaire​ MMB (Chaire Mathématique et​‌ Modélisation de la Biodiversité,​​ of Veolia-Ecole Polytechnique-Museum National​​ d'Histoire Naturelle-Fondation X).
• Mylène​​​‌ Maïda is the head‌ of the axe MEGA‌​‌ of the Réseau Thématique​​ Mathématiques et Physique (RT2173)​​​‌ funded by CNRS Mathématiques‌ (primary) and CNRS Physics‌​‌ (secondary).
• Viet Chi Tran​​ is in the steering​​​‌ committee of the Réseau‌ Thématique Matrisk (RT2167) funded‌​‌ by CNRS Mathématiques.

8.1 Promoting scientific​​​‌ activities

8.2​ Teaching - Supervision -​‌ Juries - Educational and​​ pedagogical outreach

8.3 Popularization​​​‌

9.1 Major publications​​

• 1 articleV.Vincent​​​‌ Calvez, B.Benoît‌ Henry, S.Sylvie‌​‌ Méléard and V. C.​​Viet Chi Tran.​​​‌ Dynamics of lineages in‌ adaptation to a gradual‌​‌ environmental change.Annales​​ Henri Lebesgue5July​​​‌ 2022, 729-777HAL‌DOIback to text‌​‌
• 2 articleD.Djalil​​ Chafai, A.Adrien​​​‌ Hardy and M.Mylène‌ Maïda. Concentration for‌​‌ Coulomb gases and Coulomb​​ transport inequalities.Journal​​​‌ of Functional Analysis275‌16September 2018,‌​‌ 1447-1483HALDOIback​​ to text
• 3 article​​​‌D.David Corlin Marchand‌, D.David Coupier‌​‌ and B.Benoît Henry​​. Line-of-sight Cox percolation​​​‌ on Poisson-Delaunay triangulation.‌Stochastic Processes and their‌​‌ Applications2024HALback​​ to textback to​​​‌ textback to text‌
• 4 articleD.David‌​‌ Coupier, L.Lucas​​​‌ Flammant and V. C.​Viet Chi Tran.​‌ Hyperbolic Radial Spanning Tree​​.Stochastic Processes and​​​‌ their Applications172August​ 2023, 104318HAL​‌DOIback to text​​back to text
• 5​​​‌ miscD.David Coupier​, B.Benoît Henry​‌, B.Benedikt Jahnel​​ and J.Jonas Köppl​​​‌. The Planar Lattice​ Two-Neighbor Graph Percolates.​‌December 2024HALback​​ to text
• 6 article​​​‌D.David Coupier,​ K.Kumarjit Saha,​‌ A.Anish Sarkar and​​ V. C.Viet Chi​​​‌ Tran. The 2d-directed​ spanning forest converges to​‌ the Brownian web.​​The Annals of Probability​​​‌4912021,​ 435-484HALDOIback​‌ to textback to​​ textback to text​​​‌
• 7 articleL.Laurent​ Decreusefond, J.-S.Jean-Stephane​‌ Dhersin, P.Pascal​​ Moyal and V. C.​​​‌Viet Chi Tran.​ Large graph limit for​‌ an SIR process in​​ random network with heterogeneous​​​‌ connectivity.The Annals​ of Applied Probability22​‌22012, 541-575​​HALDOIback to​​​‌ text
• 8 miscD.​David Dereudre, D.​‌Daniela Flimmel, M.​​Martin Huesmann and T.​​​‌Thomas Leblé. (Non)-hyperuniformity​ of perturbed lattices.​‌2024HALDOIback​​ to text
• 9 article​​​‌B.Benoît Henry,​ S.Sylvie Méléard and​‌ V.Viet Chi Tran​​. Time reversal of​​​‌ spinal processes for linear​ and non-linear branching processes​‌ near stationarity.Electronic​​ Journal of Probability28​​​‌noneJanuary 2023HAL​DOIback to text​‌

9.2 Publications of the​​ year

9.3‌ Cited publications

• 26 article‌​‌I.I. Akjouj,​​ M.M. Barbier,​​​‌ M.M. Clénet,‌ W.W. Hachem,‌​‌ M.M. Ma\"ida,​​ F.F. Massol,​​​‌ J.J. Najim and‌ V.V.C. Tran.‌​‌ Complex systems in Ecology:​​ A guided tour with​​​‌ large Lotka-Volterra models and‌ random matrices.Proceedings‌​‌ of the Royal Society​​ A480202302842024​​​‌back to text
• 27‌ articleF.F. Baccelli‌​‌ and C.C. Bordenave​​. The radial spanning​​​‌ tree of a Poisson‌ point process.Annals‌​‌ of Applied Probability17​​12007, 305--359​​​‌back to textback‌ to text
• 28 article‌​‌R.R. Bardenet and​​ A.A. Hardy.​​​‌ Monte Carlo with determinantal‌ point processes.Ann.‌​‌ Appl. Probab.301​​2020, 368--417DOI​​​‌back to text
• 29‌ articleN.N. Brosse‌​‌, A.A. Durmus​​​‌, E.E. Moulines​ and S.S. Sabanis​‌. The tamed unadjusted​​ Langevin algorithm.Stochastic​​​‌ Processes and their Applications​129102019,​‌ 3638--3663back to text​​
• 30 articleD.D.​​​‌ Chafaï and G.G.​ Ferré. Simulating Coulomb​‌ and log-gases with hybrid​​ Monte Carlo algorithms.​​​‌Journal of Statistical Physics​17432019,​‌ 692--714back to text​​
• 31 articleS.S.​​​‌ Clémençon, H. D.​H. De Arazoza,​‌ F.F. Rossi and​​ V.V.C. Tran.​​​‌ A statistical network analysis​ of the HIV/AIDS epidemics​‌ in Cuba.Social​​ Network Analysis and Mining​​​‌52015, Art.58​back to text
• 32​‌ unpublishedI.I. Condamine-Ducreux​​, C.C. Dumont​​​‌, V.V.C. Tran​, A.A. Cousien​‌, F.F. Barin​​, K.K. Stefic​​​‌, S.S. Deuffic-Burban​, A.A. Chollet​‌, P.P. Feuillet​​, C.C. Jangal​​​‌, M. L.M.​ Le Breton, G.​‌G. Brodsky, J.-S.​​J.-S. Dhersin, Y.​​​‌Y. Yazdanpanah and M.​M. Jauffret-Roustide. Social​‌ supports and networks linked​​ to HCV and HIV​​​‌ exposure among people who​ inject drugs in Paris​‌ and the Paris suburbs​​.back to text​​​‌
• 33 articleD.D.​ Coupier and C.C.​‌ Tran. The 2d-directed​​ spanning forest is almost​​​‌ surely a tree.​Random Structures & Algorithms​‌4212013,​​ 59--72back to text​​​‌
• 34 articleA.A.​ Cousien, V.V.C.​‌ Tran, S.S.​​ Deuffic-Burban, M.M.​​​‌ Jauffret-Roustide, J.J.S.​ Dhersin and Y.Y.​‌ Yazdanpanah. Hepatitis C​​ treatment as prevention of​​​‌ viral transmission and level-related​ morbidity in persons who​‌ inject drugs.Hepatology​​6342016,​​​‌ 1090--1101back to text​
• 35 articleD.D.​‌ Dereudre, N.N.​​ Fernez, L.L.​​​‌ Burgnies, D.D.​ Lippens and E.E.​‌ Lheurette. Poisson distributions​​ in disordered metamaterials absorbers​​​‌.Journal of Applied​ Physics1252019back​‌ to text
• 36 article​​A.A. Durmus,​​​‌ E.E. Moulines and​ E.E. Saksman.​‌ On the convergence of​​ hamiltonian monte carlo.​​​‌arXiv preprint arXiv:1705.001662017​back to text
• 37​‌ bookP.P. Flandrin​​. Explorations in time-frequency​​​‌ analysis.Cambridge University​ Press2018back to​‌ text
• 38 articleS.​​S. Ghosh and J.​​​‌J. Lebowitz. Fluctuations,​ large deviations and rigidity​‌ in hyperuniform systems: a​​ brief survey.Indian​​​‌ J. Pure Appl. Math.​4842017,​‌ 609--631URL: https://doi.org/10.1007/s13226-017-0248-1DOI​​back to text
• 39​​​‌ articleS.S. Ghosh​ and Y.Y. Peres​‌. Rigidity and tolerance​​ in point processes: Gaussian​​​‌ zeros and Ginibre eigenvalues​.Duke Math. J.​‌2017DOIback to​​ text
• 40 articleJ.​​​‌J. Ginibre. Rigorous​ lower bound on the​‌ compressibility of a classical​​ system.Phys. Lett.​​​‌ A241967,​ 223--224back to text​‌
• 41 articleC. D.​​C. D. Howard and​​​‌ C. M.C. M.​ Newman. Geodesics and​‌ spanning trees for Euclidean​​ first-passage percolation..Ann.​​ Probab.2922001​​​‌, 577--623back to‌ text
• 42 articleQ.‌​‌Q. Le Gall,​​ B.B. B\l}aszczyszyn,​​​‌ E.E. Cali and‌ T.T. En-Najjary.‌​‌ Continuum line-of-sight percolation on​​ Poisson--Voronoi tessellations.Advances​​​‌ in Applied Probability53‌22021, 510--536‌​‌back to text
• 43​​ articleJ.J. Lebowitz​​​‌. Charge fluctuations in‌ Coulomb systems.Phys.‌​‌ Rev. A271983​​, 1491--1494back to​​​‌ text
• 44 inproceedingsE.‌Eric Lheurette, R.‌​‌Roman Buisine, O.​​Olivier Vanbesien, D.​​​‌David Dereudre, L.‌Ludovic Burgnies, T.‌​‌Thibault Deletang and B.​​Benôit Cluzel. Hyperuniform​​​‌ versus Poisson Distributions in‌ Random Metasurfaces at Infrared‌​‌ Wavelengths.17th International​​ Congress on Artificial Materials​​​‌ for Novel Wave Phenomena,‌ Metamaterials 2023oralCrète,‌​‌ Greece2023, URL:​​ https://hal.science/hal-04373710back to text​​​‌
• 45 articleP.P.‌ Martin and T.T.‌​‌ Yalcin. The charge​​ fluctuations in classical Coulomb​​​‌ systems.Journal of‌ Statistical Physics.221980‌​‌, 435--463back to​​ text
• 46 articleD.​​​‌D. Ruelle. Superstable‌ interactions in classical statistical‌​‌ mechanics.Comm. Math.​​ Phys.181970,​​​‌ 127--159URL: http://projecteuclid.org/euclid.cmp/1103842505back‌ to text
• 47 article‌​‌S.S. Torquato.​​ Hyperuniform states of matter​​​‌.Physics Reports745‌2018, 1--95back‌​‌ to text
• 48 article​​F.-Y.F.-Y. Wang and​​​‌ J.-X.J.-X. Zhu.‌ Limit theorems in Wasserstein‌​‌ distance for empirical measures​​ of diffusion processes on​​​‌ Riemannian manifolds.Ann.‌ Inst. Henri Poincaré, Probab.‌​‌ Stat.5912023​​, 437--475back to​​​‌ text