I shall give a quick introduction to the model of repeated quantum interactions and their continuous-time limit. I shall also connect them to quantum trajectories. This approach to open quantum systems makes quantum noises appearing very naturally and it shall be the occasion to present these tools, which are not so commonly shared by the physicist community.

In a celebrated 1961 paper, Landauer formulated a fundamental lower bound on the energy dissipated by computation processes. Since then, there have been many attempts to formalize, generalize and contradict Landauer's analysis. The situation became even worse with the advent of quantum computing. In a recent enlightening article, Reeb and Wolf set the game into the framework of quantum statistical mechanics,  and finally gave a precise mathematical formulation of Landauer's bound. I will discuss parts of this analysis and present some extensions of it that were obtained in a joint work with V. Jaksic.

Concepts, familiar from pure states in quantum mechanics, such as "superposition" and "transition probability", are shown to be also meaningful for generic states in infinite systems, described by funnels of type I_{infty} algebras. In the physically important case of states of Connes-von Neumann type III1, these concepts also have a physically significant operational interpretation in terms of primitive observables. (Joint work with Erling Störmer)

Une carte cFK de taille $n$ est une carte aléatoire tirée parmi toutes les cartes planaires enracinées à $n$ arêtes avec une probabilité proportionnelle à la fonction de partition de la FK-percolation critique (auto-dual) sur la carte. C'est une famille de cartes aléatoire dépendant d'un paramètre $q>0$, dont le cas $q=1$ correspond à la carte planaire uniforme. En 2013 Sheffield a trouvé une bijection, dite de Hamburger-Cheeseburger, entre une carte cFK et un modèle de mot aléatoire. Dans cet exposé, nous donnons une nouvelle présentation de cette bijection. Nous construisons ensuite la limite locale de la carte cFK pour tout $q$, et étudions quelques propriétés de la limite.

Travail de mémoire de master dirigé par Jérémie Bouttier et Nicolas Curien.

A large amount of work in perturbative superstring theory is associated to certain limits of string amplitudes with massless (or very low mass) asymptotic states, in simple string backgrounds where computations are feasible. In this talk I will focus on a much less explored regime, that of amplitudes with `highly excited' asymptotic string states, and will give an overview of a formalism (based on covariant coherent state vertex operators) which is particularly efficient for explicit computations involving highly excited strings. I will discuss generic features of `all' 2-point 1-loop amplitudes, eventually focusing on a simple explicit example to illustrate the effectiveness of this new approach, while also making contact with low energy effective field theory results.

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.

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.

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.

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.

Séminaire Laurent Schwartz — EDP et applications