Le séminaire de l’équipe Probabilités et Statistique a lieu le lundi à 11h, en salle 316 du bâtiment de Métrologie B (instructions d’accès). Vous trouverez ci-dessous le planning pour l’année universitaire en cours.
Contacts : francois.bienvenu [ɑt] univ-fcomte.fr, ahmed.zaoui [ɑt] univ-fcomte.fr
Prochaine séances :
Probability on Trees and Networks, de Lyons et Peres (2016) : chapitre 2, Random Walks and Electric Networks.
(University of Vienna)
Abstract :
We will be interested in an important statistical physics model of loop-decorated planar maps (the so-called rigid loop-O(n) model on quadrangulations). After a general introduction to the model and its phase diagram, we present a convergence result for the size of these maps as the perimeter goes to infinity. Our focus will be to provide a general toolbox to address such questions, laying emphasis on connections with branching processes, which enable to leverage the recursive structure of these maps in order to derive information about their geometry. This talk is based on joint work with Élie Aïdékon and XingJian Hu (Fudan University, Shanghai).
Exposés passés :
Abstract :
Strong Gaussian approximation provides a way to construct a Wiener process with trajectories almost surely close to the ones of the given random process. Such approximation allows to obtain other limit theorems such as law of the iterated logarithm, arcsine law etc. and build consistent asymptotic variance estimates. The first theorem of this kind was proved by Strassen in 1964. The Komlós-Major-Tusnády method providing the optimal approximation rates in the i.i.d. case attracted much attention and a lot of studies have been devoted to extension of their results onto more general random systems.
Komlós-Major-Tusnády type theorems for cumulative processes will be presented in the talk and applications to some models will be given. Among them, the main emphasis will be on Lévy flights. Let ε = {εn, n ≥ 0} be independent identically distributed random vectors taking values on the unit sphere in Rk, and {Tn, n ≥ 0} (T0 = 0) be an increasing sequence of random variables not depending on ε. A Lévy flight is a continuous process X = {X(t), t ≥ 0}, starting at zero, such that its trajectories are linear on [Tn−1, Tn] and the direction of the trajectory is chosen anew at each moment Tn−1 in accordance with vector εn. Properties of such processes depend on the nature of sequence Tn. Generally this process can be neither Markov nor homogeneous. Nevertheless it is possible to provide strong approximation results as corollaries of the respective theorems proved for cumulative processes.