Cet exposé vise à donner un aperçu de quelques problèmes classiques de probabilités géométriques impliquant une structure discrète engendrée par un ensemble aléatoire de points fini ou localement fini et plongée dans un espace le plus souvent euclidien. Ces problèmes ont en commun de se formaliser simplement via une écriture intégrale. L'enjeu est ensuite de calculer ou estimer asymptotiquement cette intégrale et dans ce cadre, l'utilisation du bon changement de variables géométrique peut s'avérer cruciale. Nous évoquerons notamment des modèles de triangles aléatoires, d'enveloppes convexes aléatoires et plus généralement de polytopes aléatoires
Je présenterai les propriétés de deux modèles d'espaces métriques aléatoires construits à partir de processus de Poisson de routes. Le premier, dans R^d, a été introduit par Aldous et Kendall il y a quelques années. On verra que l'espace métrique aléatoire obtenu est presque sûrement homéomorphe à R^d, mais que sa dimension de Hausdorff est une constante strictement plus grande que d. Pour le deuxième modèle, dans l'arbre 3-régulier, on observera une transition de phase pour le phénomène d'explosion : en fonction du paramètre qui régit les limitations de vitesse des routes, il possible ou non de rouler jusqu'à l'infini en temps fini.
Maps come with different shapes, such as trees or triangulations with many more edges. Many classes of maps have been enumerated (2-connected maps, trees, quadrangulations...), notably by Tutte, and a phenomenon of universality has been demonstrated: for the majority of them, the number of elements of size n in the class has an asymptotic of the form k 1/(r^n n^(5/2)), for a certain k and a certain r. Nevertheless, there are classes of degenerate'' maps whose behaviour is similar to that of trees, and whose number of elements of size n has an asymptotic of the form k/(r^n n^(3/2)), as for example outerplanar maps. This dichotomy of behaviour is not only observed for enumeration, but also for metrics. Indeed, in thetree'' case, the distance between two random vertices is in n^(1/2), against n^{1/4} for uniform planar maps of size $n$. This work focuses on what happens between these two very different regimes. We highlight a model depending on a parameter u>0 which exhibits the expected behaviours, and a transition between the two: depending on the position of $u$ with respect to u_C, the behaviour is that of one or the other universality class. More precisely, we observe a subcritical'' regime where the scale limit of the maps is the Brownian map, asupercritical'' regime where it is the Brownian tree and finally a critical regime where it is the 3/2-stable tree. The results are obtained using a robust method, which can be used to study a variety of similar models
We are interested in studying the behavior of geometric invariants of hyperbolic 3-manifolds, such as the length of their geodesics. A way to do so is by using probabilistic methods. That is, we consider a set of hyperbolic manifolds, put a probability measure on it, and ask what is the probability that a random manifold has a certain property. There are several models of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will present a result concerning the length spectrum -the set of lengths of all closed geodesics- of a 3-manifold constructed under this model