14 et 15 septembre 2017 - Université de Rouen



Planning

Jeudi 14 septembre

10h45 : Ouverture officielle
11h00 - 12h30 : Christophe BAHADORAN - Large deviation functionals for scalar conservation laws (Part I).
12h30 - 14h00 : Déjeuner
14h00 - 14h35 : Federico SAU - Interacting particle systems with self-duality. Hydrodynamics in dynamical random environment.
14h35 - 15h10 : Laure MARECHE - Kinetically constrained models: combinatorial results and critical timescales.
15h10 - 15h25 : Pause café
15h25 - 16h55 : Christophe BAHADORAN - Large deviation functionals for scalar conservation laws (Part II).
16h55 - 17h10 : Pause café
17h10 - 17h45 : Aurélia DESHAYES - Limite d'échelle du processus de contact sous-critique.
17h45 - 18h20 : Lucas GERIN - The Hammersley process: in and out of equilibrium.

Vendredi 15 septembre

9h00 - 10h30 : Christophe BAHADORAN - Large deviation functionals for scalar conservation laws (Part III).
10h30 - 10h45 : Pause café
10h45 - 11h20 : Paul LEMIRE - Métastabilité du processus de Blume-Capel.
11h20 - 11h55 : Arnaud Le NY - Mesures de Gibbs vs. g-measures.
12h00 - 13h45 : Déjeuner
13h45 - 14h20 : Max FATHI - Convergence of gradient flows and hydrodynamic limits: the discrete case.
14h20 - 14h55 : Thimothée THIERY - Midpoint distribution of directed polymers in 1+1D random media.
14h55 - 15h10 : Pause café
15h10 - 16h45 : Kirone MALLICK - Large Deviations of a Tracer in the Symmetric Exclusion Process.



Résumés des exposés

Mini-cours

Christophe BAHADORAN (Université Clermont Auvergne) Large deviation functionals for scalar conservation laws. In these lectures, I will discuss the structure of singular action functionals and quasi-potentials expected to arise from models, like the open TASEP, which can be viewed as small stochastic perturbations of scalar conservation laws with Bardos-Leroux-Nédélec boundary conditions BLN boundary conditions are vanishing viscosity limits of Dirichlet conditions but, due to boundary layers, they are no longer Dirichlet conditions.
Boundary large deviation costs measure the "distance" between the deviating boundary value of a large deviation "non-solution" and a set of admissible boundary values for the expected solution.

Conférences

Aurélia DESHAYES (LPMA, Université Paris Diderot) Limite d'échelle du processus de contact sous-critique. Dans cet exposé je parlerai du processus de contact sous critique. Ce processus modélise une infection qui s'éteint presque sûrement si on commence avec un nombre fini de particules infectées et qu'on choisit un paramètre d'infection suffisamment petit. Mais que se passe-t-il si, dans ce même régime, on part cette fois d'un nombre infini de particules infectées? Je présenterai un travail en collaboration avec Leonardo T. Rolla où nous donnons une description des configurations asymptotiques. Une configuration sera décrite par la collection des "emplacements macroscopiques" des zones de particules infectées et par la position relative de ces particules dans chaque zone (faisant intervenir la distribution quasi stationnaire du processus de contact modulo translation).
Ce travail est une extension d'un résultat de Andjel, Ezanno, Groisman et Rolla qui décrit le processus de contact sous critique vu depuis la particule la plus à droite en dimension 1.

Max FATHI (CNRS, Université Paul Sabatier, Toulouse) Convergence of gradient flows and hydrodynamic limits: the discrete case. Using optimal transport theory, it is possible to reformulate the equation governing the evolution in time of the law of a reversible Markov process as a gradient flow in the space of probability measures. In this talk, I will explain how this concept can be used to prove convergence of interacting particle systems to their hydrodynamic, focusing on the example of the symmetric exclusion process. Joint work with Marielle Simon.

Lucas GERIN (Ecole Polytechnique, Palaiseau) The Hammersley process: in and out of equilibrium. The Hammersley process has been introduced in the 1970's to solve a problem in Combinatorial Optimization. We will show in this talk that the equilibrium behaviour of the Hammersley process gives new insights on this "old" problem. On the other hand, we will briey discuss why the Hammersley process is also interesting in Statistical physics, particularly because its out-of-equilibrium behaviour. (based on joint works with A.-L. Basdevant, N.Enriquez, J.B.Gouéré).

Paul LEMIRE (Université de Rouen) Métastabilité du processus de Blume-Capel. La métastabilité est un état de stabilité cinétique et d'instabilité thermodynamique observé dans la nature. On présentera d'abord vulgairement ce phénomène, puis on s'intéressera rapidement à l'approche développée par J. Beltran et C. Landim. On introduira ensuite le modèle de Blume-Capel sur un tore fini dans une dynamique de Glauber, plus particulièrement à son comportement métastable lorsque la température tend vers zéro. Sur une grande échelle de temps, trois états métastables subsistent. On parlera alors de l'estimation précise du temps de transition d'un état métastable à un autre. Enfin, on discutera du cas où le tore grandit quand la température décroit, et des principales difficultés que cette hypothèse engendre.

Arnaud Le NY (EURANDOM, Eindhoven et Université Paris Est) Mesures de Gibbs vs. g-measures. Au cours de cet exposé, nous illustrerons les différences entre les notions de mesures de Gibbs et de g-mesures à travers l'exemple de modèles d'Ising à longues portées, essentiellement pour des interactions à décroissances polynomiales suffisamment lentes pour que des transitions de phases soient possibles à basse température. Nous rappelerons les notions de propriétés de Markov globales et locales en dimensions supérieures, que nous illustrerons à l'aide des interfaces et des états de Dobrushin.

Kirone MALLICK (CEA, Saclay) Large Deviations of a Tracer in the Symmetric Exclusion Process. We derive the exact formula for the large deviations of a tracer in the one dimensional symmetric simple exclusion process. This formula yields all the cumulants of the tracer position in the long time limit. Our results are valid for a system prepared out of equilibrium, with a step density profile, the tracer being initially located at the boundary. The uctuations of the tracer's position at equilibrium, when the density is uniform, are obtained as an important special case. The solution is obtained using the powerful techniques of integrable probabilities, developed recently to solve the one-dimensional Kardar-Parisi-Zhang equation.
Joint work with Takashi Imamura (Chiba) and Tomohiro Sasamoto (Tokyo).

Laure MARECHE (LPMA, Université Paris Diderot) Kinetically constrained models: combinatorial results and critical timescales. Kinetically constrained models were introduced by physicists to describe the liquid/glass transition. They are spin models on a graph in which each site tries at rate one to replace its spin with a p-coin toss, but this replacement can be done only if there are "enough'' spins at zero around the site. When p tends to zero, there are less zeroes hence less replacements; the dynamics becomes slower, which leads to divergent timescales. We are interested in the speed of that divergence. We will present a combinatorial result that yields a bound on those timescales.

Federico SAU (TU Delft, Netherlands) Interacting particle systems with self-duality. Hydrodynamics in dynamical random environment. In this talk, a class of interacting particle systems with the property of being self-dual is presented, the most popular example being the symmetric simple exclusion process. This self-duality property, which we interpret as the property of recasting an n-point correlation function in terms of n "dual" particles, holds more generally on the d-dimensional integer lattice with space-time-dependent conductances. Building up on self-duality and a "ladder" graphical construction for these particle systems, the interaction among particles turns to a negligible contribution. Therefore, the introduction of a suitable randomness for the conductances leads to a quenched hydrodynamic limit for these systems, readily derived from a quenched invariance principle of a (single) random walk in the same dynamic random environment.
Joint work (in progress) with Francesca Collet, Frank Redig and Ellen Saada.

Thimothée THIERY (Institute for Theoretical Physics, Leuven) Midpoint distribution of directed polymers in 1+1D random media. The 1+1D directed polymer (DP) problem, i.e. the equilibrium statistical mechanics of directed paths in a 2-dimensional random environment, is a classic example of disordered systems. DP models are included in the KPZ universality class and as such are connected with out-of-equilibrium statistical mechanics models such as some surface growth models and interacting particles systems, in particular the WASEP (weakly asymmetric simple exclusion process), or the stochastic Burgers equation. At large scale DP models exhibit universal properties and universal distributions of uctuations linked with the so-called KPZ fixed point and the Airy process, and characterizing precisely these distributions has triggered an important research effort in the recent years. In this talk I will show how one can obtain an explicit expression for the midpoint distribution of DPs in the stationary regime, or equivalently the distribution of the position of the maximum of the Airy process minus a parabola and a Brownian motion. This is done by relating this midpoint distribution in the case of the continuum DP to a linear response function for the stochastic Burgers equation, that is in turn related through a uctuation-dissipation relation to the two-points correlation function of the stochastic Burgers field, known exactly from a previous Bethe ansatz calculation of Imamura and Sasamoto (2013). The talk will be based on a joint work with C. Maes. arXiv:1704.06909.