Dans ce travail, nous nous sommes intéressés à l'étude asymptotique d'objets combinatoires aléatoires. Deux thèmes ont particulièrement retenu notre attention : les cartes planaires aléatoires et les modèles combinatoires liés à la théorie des fragmentations. La théorie mathématique des cartes planaires aléatoires est née à l'aube de notre millénaire avec les travaux pionniers de Benjamini & Schramm, Angel & Schramm et Chassaing & Schaeffer. Elle a ensuite beaucoup progressé, mais à l'heure où ces lignes sont écrites, de nombreux problèmes fondamentaux restent ouverts. Résumons en quelques mots clés nos principales contributions dans le domaine : l'introduction et l'étude du cactus brownien (avec J.F. Le Gall et G. Miermont), l'étude de la quadrangulation infinie uniforme vue de l'infini (avec L. Ménard et G. Miermont), ainsi que des travaux plus théoriques sur les graphes aléatoires stationnaires d'une part et les graphes empilables dans $\R^d$ d'autre part (avec I. Benjamini). La théorie des fragmentations est beaucoup plus ancienne et remonte à des travaux de Kolmogorov (1941) et de Filippov (1961). Elle est maintenant bien développée (voir par exemple l'excellent livre de J. Bertoin), et nous ne nous sommes pas focalisés sur cette théorie mais plutôt sur ses applications à des modèles combinatoires. Elle s'avère en effet très utile pour étudier différents modèles de triangulations récursives du disque (travail effectué avec J.F. Le Gall) et les recherches partielles dans les quadtrees (travail effectué avec A. Joseph). / The subject of this thesis is the asymptotic study of large random combinatorial objects. This is obviously very broad, and we focused particularly on two themes: random planar maps and their limits, and combinatorial models that are in a way linked to fragmentation theory. The mathematical theory of random planar maps is quite young and was triggered by works of Benjamini & Schramm, Angel & Schramm and Chassaing & Schaeffer. This fascinating field is still growing and fundamental problems remain unsolved. We present some new results in both the scaling limit and local limit theories by introducing and studying the Brownian Cactus (with J.F. Le Gall and G. Miermont), giving a new view point, a view from infinity, at the Uniform Infinite Planar Quadrangulation (UIPQ) and bringing more theoretical contributions on stationary random graphs and sphere packable graphs (with I. Benjamini). Fragmentation theory is much older and can be tracked back to Kolmogorov and Filippov. Our goal was not to give a new abstract contribution to this well-developed theory (see the beautiful book of J. Bertoin) but rather to apply it to random combinatorial objects. Indeed, fragmentation theory turned out to be useful in the study of the so-called random recursive triangulations of the disk (joint work with J.F. Le Gall) and partial match queries in random quadtrees (joint work with A. Joseph).

<p dir="ltr">On-orbit breakup events threaten the sustainability of space operations -- many of which humans rely on for everyday subsistence on Earth -- and hinder our ability to expand human presence deeper into space. The continuous influx of objects into orbit without sufficient mechanisms for debris removal contributes to an imbalance of sources and sinks within the volume of interest of space, intensifying orbital hazards in valuable orbits. Near-Earth fragmentation analysis methods have been developed over the course of decades, yet over 23% of all breakups over the past 10 years have, to date, unknown causes. For many additional cases, breakup causes are only partially understood. Furthermore, observation data used to decipher the causes of any fragmentation event inherently contain uncertainty, stemming from, for example, orbit determination and measurement errors. This research aims to address the role of uncertainty in near-Earth fragmentation analysis through a hybrid application of the unscented transformation technique to the forensic investigation of three unclassified Atlas V Centaur upper stage breakup events in 2018 and 2019. </p><p dir="ltr">While advancements in near-Earth space situational awareness protocol are still ongoing, the aerospace community has now set its sights farther afield, in an entirely different and more complex regime: cislunar space. With heightened international interest in support of a long-lasting presence in the vicinity of the Moon, cislunar space debris has already begun to follow. How a single fragmentation plays out is highly sensitive to slight changes in initial condition in the chaotic cislunar domain. This drives the need for appropriate debris characterization tools and a detailed dynamical understanding of the region. In response to the challenges presented, this investigation evaluates the nature of cislunar debris evolution under various initial conditions through exploitation of fundamental dynamical behaviour and structures in the neighbourhood of prominent cislunar orbits. The work finds that there is large sensitivity of resulting fragment motion to the orbit and location of origination, but there are some general features and trends that can aid in providing insights to owner-operators and the development of debris mitigation guidelines.</p>