Programme de la journée 2018
Les résumés des exposés de la journée
10h50--11h10 Benjamin Dadoun (UZH)
Self-similar growth-fragmentations as scaling limits of Markov branching processes
We provide explicit conditions, in terms of the transition kernel of its driving particle, for a Markov branching process to admit a scaling limit toward a self-similar growth-fragmentation with negative index. We also derive a scaling limit for the genealogical embedding considered as a compact real tree.
11h10--11h30 Delphin Sénizergues (Université Paris 13)
Scaling limits for graphs constructed through a generalised Rémy algorithm
We study a generalised version of Rémy's algorithm, which generates a sequence of random graphs in an iterative manner. We will show that the obtained sequence of graphs appropriately rescaled, seen as compact measured metric spaces, converges almost surely to a limit, in the so-called Gromov-Hausdorff-Prokhorov topology. The limiting object can be described as an iterative gluing of metric spaces, which generalises Aldous' stick-breaking construction of its Continuum Random Tree.
11h30--11h50 Raphaël Ducatez (Université Paris-Dauphine)
Produit de matrices aléatoires et le modèle d'Anderson 1d
Nous présentons quelques résultats généraux concernant les produit de matrices aléatoires. Sous des hypothèses assez générales, il est possible de montrer que la norme se comporte de la même manière qu'une somme de variable iid et que l'on a un théorème de type loi forte des grands nombres, théorème centrale limite ou la convergence vers le mouvement brownien. Nous appliquerons ensuite ces résultats au modèle d'Anderson à une dimension qui fut introduit en physique comme modèle permettant de décrire un électrons dans un conducteur ayant des impuretés. La conductivité du matériau dépend de la forme particulière des vecteurs propres du Hamiltonien que l'on peut retrouver avec un produit de matrices aléatoires.
11h50--12h10 Nathan Noiry (Universite Paris-Nanterre)
Random Matrices: Beyond Wigner and Marchenko-Pastur Laws
What is the behavior of a large random matrix? As it turns out, the spectrum of a large symmetric matrix with i.d.d. entries having a second moment is approaching a deterministic measure, which does not depend on the law of the entries. The first proof of this result was given by Wigner in 1958. The same universality phenomenon holds for random covariance matrices, as proved by Marchenko and Pastur in 1967. In this talk, I will recall these two results and discuss a recent generalization in the covariance case. This extension allows us to obtain a new result concerning the spectrum of a large bipartite Erdös-Rényi random graph.
13h50--14h10 Pierre Rousselin (Université Paris 13)
Chute de dimension pour les marches aléatoires sur les arbres aléatoires
Nous nous intéressons à différents modèles d'arbres aléatoires et aux marches aléatoires sur les sommets de tels arbres. Dans le cas où la marche aléatoire est transiente, la marche part presque sûrement vers l'infini en empruntant un rayon aléatoire. La loi de ce rayon est appelée la mesure harmonique sur le bord de l'arbre. Un phénomène de chute de dimension se produit : cette mesure harmonique est presque sûrement concentrée sur une partie petite (au sens de la dimension de Hausdorff) du bord de l'arbre. Autrement dit, avec grande probabilité, les trajectoires de la marche aléatoires sont presque sûrement comprises dans un sous-arbre beaucoup plus fin que l'arbre original. Cette théorie a été initiée par Russel Lyons, Robin Pemantle et Yuval Peres dans les années 1990. Plus récemment, Nicolas Curien, Jean-François Le Gall, puis Shen Lin ont étudié ce phénomène sur un autre modèle d'arbres aléatoires. Nous rappellerons leurs résultats et discuteront des généralisations (arXiv:1708.06965 et arXiv:1711.07920) sur lesquelles nous avons travaillé.
14h10--14h30 Alexandre Zhou (École des ponts)
Existence to a calibrated regime-switching local volatility model
By Gyongy’s theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labordère (2011), provide an efficient calibration procedure. But so far, no global existence result is available for the limiting SDE. We obtain existence in the special case of the LSV model called regime switching local volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. This is a joint work with Benjamin Jourdain.
14h30--14h50 Wen Sun (Inria)
On the asymptotic distribution of nucleation times of polymerization processes
We investigate a stochastic model describing the time evolution of a polymerization process. A polymer is a macromolecule resulting from the aggregation of several elementary subunits called monomers. Polymers can grow by addition of monomers or can be split into several polymers. The initial state of the system consists of isolated monomers. We study the lag time of the polymerization process, that is, the first instant when the fraction of monomers used in polymers is above some threshold. The mathematical model includes a nucleation property: If nc is defined as the size of the nucleus, polymers with a size smaller than nc are quickly fragmented into smaller polymers. For polymers of size greater than nc, the fragmentation still occurs but at a smaller rate. A scaling approach is used, by taking the volume N of the system as a scaling parameter. If nc≥3 and under quite general assumptions on the way polymers are fragmented, if T is the instant of creation of the first “stable” polymer, i.e. a polymer of size nc, then the convergence in law of T, properly normalized, is proved. We also show that, if nc ≥4, then the lag time has the same order of magnitude as T and, if nc=3, it is of the order of log(N). An original feature proved for this model is the significant variability of T. This is a well known phenomenon observed in the experiments in biology but the previous mathematical models used up to now did not exhibit this magnitude of variability. The results are proved via a series of technical estimates for occupations measures on fast time scales. Stochastic calculus with Poisson processes, coupling arguments and branching processes are the main ingredients of the proofs.
14h50 -- 15h10 Grégoire Ferré (École des ponts)
About the long time stability of Feynman-Kac models
Markov chains are a widely spread tool in physics and statistics, and studying their long time behavior is crucial for two reasons. First, it provides insight in the physics of the model, in particular its stationary properties. Second, it is a useful tool for designing stable numerical schemes for sampling high dimensional integrals. Surprisingly, such a study over the long time can be carried over through the one time step evolution operator of the dynamics, a fact known for while, in particular since the works of Meyn and Tweedie in the 90's. A key notion is that of Lyapunov function, which means, in a physical perspective, that there is some energy that decreases in average along the dynamics.
Feynman-Kac models are a generalization of Markov chains that, in addition to the dynamics itself, weight the trajectories along paths. They have been used in physics since the 70's, in which case trajectories going through regions of high energy end up with a small weight. On the other hand, it is a classical tool in the simulation of rare events: trajectories with unlikely paths are given an important weight, which results in importance sampling algorithms. As for Markov chains, the long time behavior of these dynamics is crucial for understanding the properties of the system and designing reasonable numerical schemes. Surprisingly, few results were known for unbounded state spaces so far, which is the typical situation in reality. This talk will be a presentation a work done in collaboration with M. Rousset and G. Stoltz on this stability issue. In particular, the notion of Lyapunov function is extended for such systems with weighted trajectories, which results in criteria that are easy to check in practice.
15h10--15h30 Ronan Herry (Université Paris 5 et Université du Luxembourg)
On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group
Gross proved the logarithmic Sobolev inequality for the Gaussian measure via a tensorization property and the central limit theorem. This methodology can be adapted to the Heisenberg group to produce a logarithmic Sobolev inequality for the heat kernel. I will explain the methodology and the importance of understanding logarithmic Sobolev inequalities on the Heisenberg group, that is the simplest non-trivial instance of a sub-Riemannian manifold.
15h30--15h50 Paul Dario (Universite Paris-Dauphine et ENS)
Homogénéisation stochastique et percolation surcritique
La théorie de l'homogénéisation stochastique telle que développée par Armstrong, Kuusi et Mourrat requiert une condition d'uniforme ellipticité sur l'environnement. La question de l'adaptabilité de cette théorie dans le cas d'un milieu poreux (lorsque l'on affaiblit l'hypothèse d'uniforme ellipticité) se pose alors. Nous répondons à cette question dans la cas de la percolation sur-critique en développant un théorie de l'homogénéisation sur l'amas infini. Pour cela nous présenterons un argument de renormalisation de l'amas infini permettant d'adapter la théorie puis nous en exposerons les principaux résultats : théorème d'homogénéisation quantitative, régularité à grande échelle et théorème de Liouville.
15h50--16h10 Thomas Lehéricy (Université Paris-Sud)
Isoperimetric inequalities in random maps
A random map is a locally finite planar graph embedded on a surface, seen up to orientation-preserving homeomorphisms. They have attracted considerable attention since the eighties in the context of quantum gravity.
We are interested in geometric property of the uniform infinite quadrangulation of the plane. Unlike in regular lattices like the d-dimentional grid, there can be very large sets with a very small perimeter. We study the perimeter of sets containing a ball around the origin, and derive an isoperimetric inequality in this model.