Publications

2025

• Benchmarking Powell's Legacy: Performance of Five Derivative-Free Solvers in pdfo on the bbob Test Suite
  
  • Brockhoff Dimo
  • Villain Tanguy
  , 2025, pp.1833-1841. The pdfo library by Tom M. Ragonneau and Zaikun Zhang makes the five derivative-free solvers BOBYQA, COBYLA, LINCOA, NEWUOA, and UOBYQA—originally written by Michael J. D. Powell—available in Python. In this paper, we are comparing their performance on the bbob test suite with three other solvers from the COCO data archive: CMA-ES from pycma, SLSQP and BFGS from scipy. We also compare the original solvers, written by Powell in Fortran 77, with the current pdfo versions, which saw multiple bug fixes and code improvements by Ragonneau and Zhang. For the latter comparison, we do not see large effects on performance between the Fortran 77 version and the current pdfo version. The only notable exception is the Bent Cigar function where we observe differences by a factor of 2–5 for BOBYQA, LINCOA, and NEWUOA. Compared to the other baseline algorithms, BOBYQA, LINCOA and NEWUOA perform very similarly over all bbob functions, being about a factor of 5 slower than SLSQP and BFGS while UOBYQA—as the best-performing pdfo solver—outperforms SLSQP and BFGS for larger budgets when compared over all 24 bbob functions. The linear surrogate of COBYLA, on the contrary, is clearly worse over all functions than the other algorithms. (10.1145/3712255.3734343)
  DOI : 10.1145/3712255.3734343
• On the Pareto Set and Front of Multiobjective Spherical Functions with Convex Constraints
  
  • Auger Anne
  • Brockhoff Dimo
  • Cork Jordan
  • Tušar Tea
  , 2025, pp.527-535. We analyze a fundamental class of multiobjective constrained problems where the objectives are spherical functions and the constraints are convex. As an application from the projection theorem on closed convex sets, we prove that the constrained Pareto set corresponds to the orthogonal projection of the unconstrained Pareto set onto the feasible region. We establish this fundamental geometric property and illustrate its implications using visualizations of Pareto sets and fronts under various constraint configurations. Furthermore, we assess the performance of NSGA-II on these problems, examining its ability to approximate the constrained Pareto set across different dimensions. Our findings highlight the importance of theoretically grounded and understood benchmark problems for assessing algorithmic behavior and contribute to a deeper understanding of constrained multiobjective landscapes. (10.1145/3712256.3726432)
  DOI : 10.1145/3712256.3726432
• How Robust is UOBYQA to Worsening, Frozen Noise? Investigations on the bbob Test Suite With Outliers
  
  • Brockhoff Dimo
  • Villain Tanguy
  , 2025, pp.1842 - 1849. UOBYQA, short for Unconstrained Optimization By Quadratic Approximation, is one of the well-known solvers derived and implemented by Michael J. D. Powell. In each step, the algorithm builds a quadratic surrogate of the objective function, interpolating quadratically many points for which the true function values are known. The model is optimized within the so-called trust region and the resulting solution is evaluated next. Adaptation of the trust region radius allows for fast convergence on a wide range of (noiseless) functions without the need for derivatives. In this workshop paper, we investigate the effect of (frozen) nonnegative, i.e., worsening noise on UOBYQA with varying probability of solutions being affected by the noise. To this end, we use the COCO platform and its newest addition, the noiser, applied to the classical bbob functions. The numerical benchmarking experiments showcase that UOBYQA is negatively affected by the noise, but surprisingly little over a wide range of noise strengths for some of the bbob functions. (10.1145/3712255.3734354)
  DOI : 10.1145/3712255.3734354
• Rank-based Linear-Quadratic Surrogate Assisted CMA-ES
  
  • Gharafi Mohamed
  • Hansen Nikolaus
  • Le Riche Rodolphe
  • Brockhoff Dimo
  , 2025. In this poster, we introduce a rank-based surrogate-assisted variant of CMA-ES. Unlike previous methods that employ rank information as constraints to train an SVM classifier, our approach employs a linear-quadratic regression on the ranks. We investigate the method's invariance empirically. While this first algorithm outperforms CMA-ES with a few exceptions, it falls short to entirely meet the lq-CMA-ES performance levels. To address this, we propose an enhanced variant that handles together two alternative surrogates, one based on the ranks and one based on the original function values. Although this variant sacrifices strict invariance, it gains in robustness and achieves performance comparable to, or even exceeding, lq-CMA-ES on transformed problems. This last algorithm shows how simply incorporating new transformations of rank values could improve any surrogate-based CMA-ES variant.
• Benchmarking CMA-ES under Additive and Subtractive Noise on the BBOB Testbed
  
  • Girardin Oskar
  , 2025, pp.1867-1874. We benchmark a non-elitist CMA-ES algorithm on the BBOB testbed with additive and subtractive noise. In particular, we consider the case where re-evaluated solutions produce the same observed function value. As a comparison, we benchmark a version of CMA-ES with resampling, which aims at reducing the effective noise level. We find CMA-ES to be more sensitive to subtractive noise than to additive noise in dimensions 2, 3, 5, 10, 20 and 40. Resampling for CMA-ES appears to be detrimental for low noise levels, while it is beneficial for high noise levels. (10.1145/3712255.3734332)
  DOI : 10.1145/3712255.3734332
• Revisiting Non-Acyclic GFlowNets in Discrete Environments
  
  • Morozov Nikita
  • Maksimov Ian
  • Tiapkin Daniil
  • Samsonov Sergey
  , 2025. Generative Flow Networks (GFlowNets) are a family of generative models that learn to sample objects from a given probability distribution, potentially known up to a normalizing constant. Instead of working in the object space, GFlowNets proceed by sampling trajectories in an appropriately constructed directed acyclic graph environment, greatly relying on the acyclicity of the graph. In our paper, we revisit the theory that relaxes the acyclicity assumption and present a simpler theoretical framework for non-acyclic GFlowNets in discrete environments. Moreover, we provide various novel theoretical insights related to training with fixed backward policies, the nature of flow functions, and connections between entropy-regularized RL and non-acyclic GFlowNets, which naturally generalize the respective concepts and theoretical results from the acyclic setting. In addition, we experimentally re-examine the concept of loss stability in non-acyclic GFlowNet training, as well as validate our own theoretical findings. (10.48550/arXiv.2502.07735)
  DOI : 10.48550/arXiv.2502.07735
• Prediction-Aware Learning in Multi-Agent Systems
  
  • Capitaine Aymeric
  • Boursier Etienne
  • Moulines Eric
  • Jordan Michael I.
  • Durmus Alain
  , 2025, PMLR 267. The framework of uncoupled online learning in multiplayer games has made significant progress in recent years. In particular, the development of time-varying games has considerably expanded its modeling capabilities. However, current regret bounds quickly become vacuous when the game undergoes significant variations over time, even when these variations are easy to predict. Intuitively, the ability of players to forecast future payoffs should lead to tighter guarantees, yet existing approaches fail to incorporate this aspect. This work aims to fill this gap by introducing a novel prediction-aware framework for time-varying games, where agents can forecast future payoffs and adapt their strategies accordingly. In this framework, payoffs depend on an underlying state of nature that agents predict in an online manner. To leverage these predictions, we propose the POWMU algorithm, a contextual extension of the optimistic Multiplicative Weight Update algorithm, for which we establish theoretical guarantees on social welfare and convergence to equilibrium. Our results demonstrate that, under bounded prediction errors, the proposed framework achieves performance comparable to the static setting. Finally, we empirically demonstrate the effectiveness of POWMU in a traffic routing experiment.
• On Teacher Hacking in Language Model Distillation
  
  • Tiapkin Daniil
  • Calandriello Daniele
  • Ferret Johan
  • Perrin Sarah
  • Vieillard Nino
  • Ramé Alexandre
  • Blondel Mathieu
  , 2025. Post-training of language models (LMs) increasingly relies on the following two stages: (i) knowledge distillation, where the LM is trained to imitate a larger teacher LM, and (ii) reinforcement learning from human feedback (RLHF), where the LM is aligned by optimizing a reward model. In the second RLHF stage, a well-known challenge is reward hacking, where the LM over-optimizes the reward model. Such phenomenon is in line with Goodhart's law and can lead to degraded performance on the true objective. In this paper, we investigate whether a similar phenomenon, that we call teacher hacking, can occur during knowledge distillation. This could arise because the teacher LM is itself an imperfect approximation of the true distribution. To study this, we propose a controlled experimental setup involving: (i) an oracle LM representing the ground-truth distribution, (ii) a teacher LM distilled from the oracle, and (iii) a student LM distilled from the teacher. Our experiments reveal the following insights. When using a fixed offline dataset for distillation, teacher hacking occurs; moreover, we can detect it by observing when the optimization process deviates from polynomial convergence laws. In contrast, employing online data generation techniques effectively mitigates teacher hacking. More precisely, we identify data diversity as the key factor in preventing hacking. Overall, our findings provide a deeper understanding of the benefits and limitations of distillation for building robust and efficient LMs.
• Finite-Sample Convergence Bounds for Trust Region Policy Optimization in Mean-Field Games
  
  • Ocello Antonio
  • Tiapkin Daniil
  • Mancini Lorenzo
  • Laurière Mathieu
  • Moulines Eric
  , 2025. We introduce Mean-Field Trust Region Policy Optimization (MF-TRPO), a novel algorithm designed to compute approximate Nash equilibria for ergodic Mean-Field Games (MFG) in finite state-action spaces. Building on the well-established performance of TRPO in the reinforcement learning (RL) setting, we extend its methodology to the MFG framework, leveraging its stability and robustness in policy optimization. Under standard assumptions in the MFG literature, we provide a rigorous analysis of MF-TRPO, establishing theoretical guarantees on its convergence. Our results cover both the exact formulation of the algorithm and its sample-based counterpart, where we derive high-probability guarantees and finite sample complexity. This work advances MFG optimization by bridging RL techniques with mean-field decision-making, offering a theoretically grounded approach to solving complex multi-agent problems. (10.48550/arXiv.2505.22781)
  DOI : 10.48550/arXiv.2505.22781
• Unified Breakdown Analysis for Byzantine Robust Gossip
  
  • Gaucher Renaud
  • Dieuleveut Aymeric
  • Hendrikx Hadrien
  , 2025, 267, pp.18868-18896. In decentralized machine learning, different devices communicate in a peer-to-peer manner to collaboratively learn from each other's data. Such approaches are vulnerable to misbehaving (or Byzantine) devices. We introduce F-RG, a general framework for building robust decentralized algorithms with guarantees arising from robust-sum-like aggregation rules F. We then investigate the notion of breakdown point, and show an upper bound on the number of adversaries that decentralized algorithms can tolerate. We introduce a practical robust aggregation rule, coined CSours, such that CSours-RG has a near-optimal breakdown. Other choices of aggregation rules lead to existing algorithms such as ClippedGossip or NNA. We give experimental evidence to validate the effectiveness of CSours-RG and highlight the gap with NNA, in particular against a novel attack tailored to decentralized communications.
• High Performance Parallel Solvers for the time-harmonic Maxwell Equations
  
  • Fressart Elise
  • Dubois Sébastien
  • Gouarin Loïc
  • Massot Marc
  • Nowak Michel
  • Spillane Nicole
  , 2025. We consider the numerical solution of large scale time-harmonic Maxwell equations. To this day, this problem remains difficult, in particular because the equations are neither Hermitian nor semi-definite. Our approach is to compare different strategies for solving this set of equations with preconditioners that are available either in PETSc, MUMPS, or in hypre. Four different preconditioners are considered. The first is the sparse approximate inverse, which is often applied to electromagnetic problems. The second is Restricted Additive Schwarz, a domain decomposition preconditioner. The third is the Hiptmair-Xu preconditioner which is tailored to the positive Maxwell equations, a nearby problem. The final preconditioner is MUMPS's Block Low-Rank method, a compressed block procedure. We also compare the performance of this method to the standard LU factorization technique, which is a direct solver. Performance with respect to the mesh size, the number of CPU cores, the wavelength and the physical size of the domain are considered. This work in progress yields temporary conclusions in favour of the Hiptmair-Xu and the Block Low-Rank preconditioners.
• Signed tropicalization of polar cones
  
  • Akian Marianne
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Sergeev Sergei
  Journal of Optimization Theory and Applications, Springer Verlag, 2025, 207 (10). We study the tropical analogue of the notion of polar of a cone, working over the semiring of tropical numbers with signs. We characterize the cones which arise as polars of sets of tropically nonnegative vectors by an invariance property with respect to a tropical analogue of Fourier-Motzkin elimination. We also relate tropical polars with images by the nonarchimedean valuation of classical polars over real closed nonarchimedean fields and show, in particular, that for semi-algebraic sets over such fields, the operation of taking the polar commutes with the operation of signed valuation (keeping track both of the nonarchimedean valuation and sign). We apply these results to characterize images by the signed valuation of classical cones of matrices, including the cones of positive semidefinite matrices, completely positive matrices, completely positive semidefinite matrices, and their polars, including the cone of co-positive matrices, showing that hierarchies of classical cones collapse under tropicalization. We finally discuss an application of these ideas to optimization with signed tropical numbers. (10.1007/s10957-025-02732-2)
  DOI : 10.1007/s10957-025-02732-2
• On Global Convergence Rates for Federated Policy Gradient under Heterogeneous Environment
  
  • Labbi Safwan
  • Mangold Paul
  • Tiapkin Daniil
  • Moulines Eric
  , 2025. Ensuring convergence of policy gradient methods in federated reinforcement learning (FRL) under environment heterogeneity remains a major challenge. In this work, we first establish that heterogeneity, perhaps counter-intuitively, can necessitate optimal policies to be non-deterministic or even time-varying, even in tabular environments. Subsequently, we prove global convergence results for federated policy gradient (FedPG) algorithms employing local updates, under a Łojasiewicz condition that holds only for each individual agent, in both entropy-regularized and non-regularized scenarios. Crucially, our theoretical analysis shows that FedPG attains linear speed-up with respect to the number of agents, a property central to efficient federated learning. Leveraging insights from our theoretical findings, we introduce b-RS-FedPG, a novel policy gradient method that employs a carefully constructed softmax-inspired parameterization coupled with an appropriate regularization scheme. We further demonstrate explicit convergence rates for b-RS-FedPG toward near-optimal stationary policies. Finally, we demonstrate that empirically both FedPG and b-RS-FedPG consistently outperform federated Q-learning on heterogeneous settings. (10.48550/arXiv.2505.23459)
  DOI : 10.48550/arXiv.2505.23459
• Large-Eddy simulation of solid/fluid heat and mass transfer applied to the thermal degradation of composite materials
  
  • Grenouilloux Adrien
  • Bioche Kévin
  • Dellinger Nicolas
  • Letournel Roxane
  • Bechane Yacine
  • Moureau Vincent
  , 2025.
• Improving GFlowNets with Monte Carlo Tree Search
  
  • Morozov Nikita
  • Tiapkin Daniil
  • Samsonov Sergey
  • Naumov Alexey
  • Vetrov Dmitry
  , 2024. Generative Flow Networks (GFlowNets) treat sampling from distributions over compositional discrete spaces as a sequential decision-making problem, training a stochastic policy to construct objects step by step. Recent studies have revealed strong connections between GFlowNets and entropy-regularized reinforcement learning. Building on these insights, we propose to enhance planning capabilities of GFlowNets by applying Monte Carlo Tree Search (MCTS). Specifically, we show how the MENTS algorithm (Xiao et al., 2019) can be adapted for GFlowNets and used during both training and inference. Our experiments demonstrate that this approach improves the sample efficiency of GFlowNet training and the generation fidelity of pre-trained GFlowNet models. (10.48550/arXiv.2406.13655)
  DOI : 10.48550/arXiv.2406.13655
• Contribution to the study of the mean field games master equation
  
  • Meynard Charles
  , 2025. This thesis focuses on the problems of uniqueness and existence to the mean field games master equation. The first part is devoted to the study of this equation for mean field games in which the optimization problem solved by an individual player depends on the realization of a stochastic process common to all players. This process may be autonomous, but it can also depend on the evolution of the game through the distribution of players. We provide sufficient conditions ensuring the existence and uniqueness of a solution to the master equation associated with such games, first in the context of finite state space mean field games. We then extend these results to master equations defined on the space of probability measures under several notions of monotonicity. In the second part, we introduce the notion of monotone solution for so-called displacement monotone mean field games. This notion of weak solution to the master equation does not require any differentiability assumptions on the solution with respect to probability measures. Under suitable monotonicity conditions, we establish uniqueness, stability, and finally existence of such solutions.
• High-order adaptive multi-domain time integration scheme for microscale lithium-ion batteries simulations
  
  • Asad Ali
  • de Loubens Romain
  • François Laurent
  • Massot Marc
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2025, 11, pp.369-404. We investigate the modeling and simulation of ionic transport and charge conservation in lithium-ion batteries (LIBs) at the microscale. It is a multiphysics problem that involves a wide range of time scales. The associated computational challenges motivate the investigation of numerical techniques that can decouple the time integration of the governing equations in the liquid electrolyte and the solid phase (active materials and current collectors). First, it is shown that semi-discretization in space of the non-dimensionalized governing equations leads to a system of index-1 semi-explicit differential algebraic equations (DAEs). Then, a new generation of strategies for multi-domain integration is presented, enabling high-order adaptive coupling of both domains in time, with efficient and potentially different domain integrators. They reach a high level of flexibility for real applications, beyond the limitations of multirate methods. A simple 1D LIB half-cell code is implemented as a demonstrator of the new strategy for the simulation of different modes of cell operation. The integration of the decoupled subsystems is performed with high-order accurate implicit nonlinear solvers. The accuracy of the space discretization is assessed by comparing the numerical results to the analytical solutions. Then, temporal convergence studies demonstrate the accuracy of the new multi-domain coupling approach. Finally, the accuracy and computational efficiency of the adaptive coupling strategy are discussed in the light of the conditioning of the decoupled subproblems compared to the one of the fully-coupled problem. This new approach will constitute a key ingredient for the high-fidelity 3D LIB simulations based on actual electrode microstructures. (10.5802/smai-jcm.128)
  DOI : 10.5802/smai-jcm.128
• Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process
  
  • Basdevant Anne-Laure
  • Gerin Lucas
  • Marivain Maxime
  , 2025. A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution.
• Exponentially Fading Memory Signature
  
  • Abi Jaber Eduardo
  • Sotnikov Dimitri
  , 2025. We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a "stationarized" representation, making it particularly well-suited for timeseries analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula à la Fawcett for its expected value in terms of Magnus expansions. We also study linear combinations of EFM-signature elements and the computation of associated characteristic functions in terms of a mean-reverting infinite dimensional Riccati equation.
• Stochastic and deterministic approaches for studying telomere shortening and other cell dynamics
  
  • Olayé Jules
  , 2025. Telomeres are non-coding regions situated at the ends of chromosomes of eukaryotic cells, whose lengths vary during cell divisions through shortening and/or elongation. The shortening of telomeres is one of the main factors leading to replicative senescence, an irreversible state in which cells stop to divide. The study of telomeres has gained interest in recent years, notably due to their link with the emergence of cancer cells and with the aging of individuals. These studies have led to the development of mathematical models, allowing the description of the experiments conducted by biologists. Using these models often relies on assuming that a stationary profile of telomere lengths exists, or doing model approximations, for which theoretical results are not well-established. The main objective of this thesis is to provide theoretical results, through two axes of study.The first axis is the analysis of the long-time behaviour of the telomere lengths density in a telomere shortening model with elongation. The difficulties that we need to manage are the non-compact nature of the operator we study, and the semi-Markovian aspect of our model. The second axis is the study of an inverse problem, in which we aim to estimate an initial distribution of telomere lengths from measurements of senescence times. The main argument in solving this problem is to approximate our models by transport and transport-diffusion equations, on which estimators are easier to construct.In parallel with these works related to telomeres, two additional studies have been conducted in this thesis, still related to cell dynamics. In the first one, we propose a model for the neurogenesis phenomenon based on compound Poisson processes. In the second one, we perform a theoretical and numerical study of a method for estimating the division times distribution of bacteria, developed by a biologist during his thesis.
• From random matrices to systems of particles in interaction
  
  • Pesce Valentin
  , 2025. The goal of these expository notes is to give an introduction to random matrices for non-specialist of this topic focusing on the link between random matrices and systems of particles in interaction. We first recall some general results about the random matrix theory that create a link between random matrices and systems of particles through the knowledge of the law of the eigenvalues of certain random matrices models. We next focus on a continuous in time approach of random matrices called the Dyson Brownian motion. We detail some general methods to study the existence of system of particles in singular interaction and the existence of a mean field limit for these systems of particles. Finally, we present the main result of large deviations when studying the eigenvalues of random matrices. This method is based on the fact that the eigenvalues of certain models of random matrices can be viewed as log gases in dimension 1 or 2.
• A quantitative comparison of high-order asymptotic-preserving and asymptotically-accurate IMEX methods for the Euler equations with non-ideal gases
  
  • Orlando Giuseppe
  • Boscarino Sebastiano
  • Russo Giovanni
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2025, 442, pp.118037. We present a quantitative comparison between two different Implicit-Explicit Runge-Kutta (IMEX-RK) approaches for the Euler equations of gas dynamics, specifically tailored for the low Mach limit. In this regime, a classical IMEX-RK approach involves an implicit coupling between the momentum and energy balance so as to avoid the acoustic CFL restriction, while the density can be treated in a fully explicit fashion. This approach leads to a mildly nonlinear equation for the pressure, which can be solved according to a fixed point procedure. An alternative strategy consists of employing a semi-implicit temporal integrator based on IMEX-RK methods (SI-IMEX-RK). The stiff dependence is carefully analyzed, so as to avoid the solution of a nonlinear equation for the pressure also for equations of state (EOS) of non-ideal gases. The spatial discretization is based on a Discontinuous Galerkin (DG) method, which naturally allows high-order accuracy. The asymptotic-preserving (AP) and the asymptotically-accurate (AA) properties of the two approaches are assessed on a number of classical benchmarks for ideal gases and on their extension to non-ideal gases. (10.1016/j.cma.2025.118037)
  DOI : 10.1016/j.cma.2025.118037
• Spatio-temporal thermalization and adiabatic cooling of guided light waves
  
  • Zanaglia Lucas
  • Garnier Josselin
  • Carusotto Iacopo
  • Doya Valérie
  • Michel Claire
  • Picozzi Antonio
  , 2025. We propose and theoretically characterize three-dimensional spatio-temporal thermalization of a continuous-wave classical light beam propagating along a multi-mode optical waveguide. By combining a non-equilibrium kinetic approach based on the wave turbulence theory and numerical simulations of the field equations, we anticipate that thermalizing scattering events are dramatically accelerated by the combination of strong transverse confinement with the continuous nature of the temporal degrees of freedom. In connection with the blackbody catastrophe, the thermalization of the classical field in the continuous temporal direction provides an intrinsic mechanism for adiabatic cooling and, then, spatial beam condensation. Our results open new avenues in the direction of a simultaneous spatial and temporal beam cleaning. (10.48550/arXiv.2506.23536)
  DOI : 10.48550/arXiv.2506.23536
• Computer-assisted proofs for differential equations and dynamical systems
  
  • Breden Maxime
  , 2025.
• On random 3/2-stable maps
  
  • Kammerer Emmanuel
  , 2025. This thesis focuses on the asymptotic behaviour of random planar maps and random trees. In a first part, we study the geometry of large random planar maps with high degree vertices, also called stable maps, with a particular interest in the critical case: the case of 3/2-stable maps. We prove that these maps appear naturally in the study of random planar maps coupled with a loop O(n) model, a classical model of statistical mechanics, in the case n=2. We obtain the asymptotic behaviour of the distances between two uniform random vertices and we prove that the diameter is of the same order. A consequence of these results is that 3/2-stable maps do not satisfy scaling limits in the usual sense of Gromov-Hausdorff. However, we get the scaling limit of the distances between high degree vertices and the root. This scaling limit corresponds to a distance between the loops of a conformal loop ensemble and the boundary of the disc, measured using an independent Liouville quantum gravity, which is the Lamperti transform of the "quantum distance" to the boundary introduced by Aru, Holden, Powell and Sun. We then study the geodesics to the root for the first passage percolation distance on stable maps. We construct the scaling limit of the geodesics by means of a coalescing flow of pure jump diffusions.In a second part, we introduce a generalisation of the model of random recursive trees by adding a freezing mechanism. At some steps, a uniform random vertex is frozen and new vertices can no longer be attached to this vertex. The infection tree in an SIR model on the complete graph falls within this framework. We obtain local limits of these trees and scaling limits of the distances between the root and a typical vertex, of the distance between two uniform typical vertices and of the total height of the tree obtained after n steps. In the sub-linear case where the number of non-frozen vertices evolves as a given power of n, we identify a phase transition. Lastly, we consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalised converge.