Programme de la journée « cartes aléatoires » du 9 décembre à l'IHES.
10h30 café d'accueil
11h Emmanuel Guitter (CEA)
La fonction à trois points des cartes planaires générales
Je montrerai comment calculer la fonction à trois points dépendant des distances pour la famille des cartes planaires générales, c'est à dire la fonction génératrice de ces cartes avec poids par arête et poids par face, avec trois sommets marqués à distances mutuelles prescrites. Je discuterai aussi du cas de la famille des cartes biparties et de quelques cas limites. Ceci est un travail en commun avec Éric Fusy.
12h Déjeuner (plateaux repas)
13h30 Gourab Ray (Cambridge)
Hyperbolic random maps: an overview
Abstract: Recently, hyperbolic versions of uniform planar maps have attracted a great deal of attention. These maps are conjectured to be local limits of uniform maps embedded on high genus surfaces. First, I will describe a resolution of this conjecture for unicellular (or one-face) maps. Although for other cases this still remains a conjecture, several possible candidates have been constructed. I will give a brief overview of these models, their construction and geometric properties. I will also discuss behaviour of random walks (e.g. their speed) on them and how the ''final behaviour" of random walks on them can be nicely described via their circle packings. Parts of these works are joint with Omer Angel, Guillaume Chapuy, Nicolas Curien, Tom Hutchcroft and Asaf Nachmias.
14h30 Ioan Manolescu (Genève)
Scaling limits and influence of the seed graph in preferential attachment trees
We investigate two aspects of large random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees. Starting with a given tree (called the seed), a random sequence of trees is built by adding vertices one by one, connecting them to one of the existing vertices chosen randomly with probability proportional to its degree. Bubeck, Mossel and Racz conjectured that the law of the trees obtained after adding a large number of vertices still carries information about the seed from which the process started. We confirm this conjecture using an observable based on the number of ways of embedding a given (small) tree in a large tree obtained by preferential attachment. Next we study scaling limits of such trees. Since the degrees of vertices of a large preferential attachment tree are much higher than its diameter, a simple scaling limit would lead to a non locally compact space that fails to capture the structure of the object. Yet, for a planar version of the model, a much more convenient limit may be defined via its loop tree. The limit is a new object called the Brownian tree, obtained from the CRT by a series of quotients.
16h00 Vincent Vargas (CNRS, Ens Ulm)
Liouville quantum gravity on the Riemann sphere
Abstract: in this talk, I will present a rigorous probabilistic construction of Liouville Field Theory on the Riemann sphere with positive cosmological constant, as considered by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings". Then, I will explain some of the fundamental properties of the theory like conformal covariance under PSL$_2(\C)$-action, Seiberg bounds, KPZ scaling laws, the KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula. If time permits, I will also explain the construction in the disk. This is based on joint works (some on arxiv and others in progress) with F. David, Y. Huang, A. Kupiainen, R. Rhodes.