Mercredi 8 juin

14h30 - 15h30 : Jean BRICMONT (UCL, Louvain). On the derivation of Fourier's law.
15h30 - 15h50 : Pause
15h50 - 17h50 : Enza ORLANDI (Rome 3). Lecture 1 : Random Field Ising Model with special enphasis on Kac type interactions.

Jeudi 9 juin

09h15 - 11h15 : Enza ORLANDI (Rome 3). Lecture 2 : Random Field Ising Model with special enphasis on Kac type interactions.
11h15 - 11h45 : Pause
11h45 - 12h45 : Raphaël LEFEVERE (Paris). Perturbative analysis of the anharmonic chain of oscillators out of equilibrium.

14h30 - 15h00 : Nicolas LANCHIER (Rouen). Stochastic spatial models of host-pathogen and host-mutualist interactions.
15h00 - 16h00 : Krishnamurthi RAVISHANKAR (SUNY - New Paltz). Convergence of the coalescing random walk lattice filling curve to the Tóth-Werner plane filling curve.
16h00 - 16h30 : Pause
16h30 - 17h30 : Jean-René CHAZOTTES (CNRS, Polytechnique). Deviation inequalities for stochastic processes and random fields. A coupling approach.
17h30 - 18h00 : Nicolas PETRELIS (Rouen). Pinning of a polymer at an interface.

Vendredi 10 juin

09h00 - 11h00 : Enza ORLANDI (Rome 3). Lecture 3 : Random Field Ising Model with special enphasis on Kac type interactions.
11h00 - 11h30 : Pause
11h30 - 12h30 : Carlangelo LIVERANI (Rome 2). Coupled Markov chains and coupled map lattices.

COURS

Enza ORLANDI (Rome 3)

Random Field Ising Model with special enphasis on Kac type interactions.

I will give an overview of known results on Ising spin systems with external random field. I will then focus on new results obtained in collaboration with Pierre Picco about the characterization of the typical configurations for one-dimensional random field Kac model. The last part of the lectures will be devoted to study the dynamic of particles systems associated to such models.

CONFÉRENCES

Jean BRICMONT (UCL, Louvain)

On the derivation of Fourier's law.

We consider a lattice of coupled anharmonic oscillators in contact, on their boundaries, with heat baths at two different temperatures. We write approximate equations for the correlation functions of the stationary state of that system, obtained after "closing" the true equations in a way similar to what Boltzmann did for dilute gases. Within this approximation, we show that the stationary state exhibits a temperature profile satisfying Fourier's law.

Jean-René CHAZOTTES (CNRS, Polytechnique)

Deviation inequalities for stochastic processes and random fields. A coupling approach.

I present a simple approach to obtain deviation inequalities for non-product measures based on coupling. In the case of random fields,this provides upper bounds for the probability of deviation of an arbitrary local function from its expected value. I shall illustrate these results with the Ising model. For the high-temperature Ising model, this gives an exponential bound while in the low-temperature case, this gives a polynomial bound I will discuss some applications of these deviation inequalities This is a joint work with Pierre Collet, Christoff Kuelske and Frank Redig.

Nicolas LANCHIER (Université de Rouen)

Stochastic spatial models of host-pathogen and host-mutualist interactions.

Mutualists and pathogens, collectively called symbionts, are ubiquitous in plant communities. While some symbionts are highly host-specific, others associate with multiple hosts. The outcome of multispecies host-symbionts interactions with different degrees of specificity are difficult to predict at this point due to a lack of a general conceptual framework. Complicating our predictive power is the fact that plant populations are spatially explicit and we know from past research that explicit space can profoundly alter plant-plant interactions. We introduce a spatially explicit, stochastic model to investigate the role of explicit space and host-specificity in multispecies host-symbiont interactions. We find that in our model, pathogens can significantly alter the spatial structure of plant communities, promoting coexistence, whereas mutualists appear to have only a limited effect. Effects are more pronounced the more host-specific symbionts are. This is a joint work with Claudia Neuhauser.

Raphaël LEFEVERE (Paris)

Perturbative analysis of the anharmonic chain of oscillators out of equilibrium.

We compute the first-order correction to the correlation functions of the stationary state of a stochastically forced harmonic chain out of equilibrium when a small on-site anharmonic potential is added.

Carlangelo LIVERANI (Rome 2)

Coupled Markov chains and coupled map lattices.

I will illustrate a simple idea to study the statistical properties of weakly coupled mixing systems.

Nicolas PETRELIS (Université de Rouen)

Pinning of a polymer at an interface.

We consider a model of hydrophobic homopolymer in interaction with an interface between oil and water. This polymer gains random prices each time it crosses the interface, so it tries to stay in the neighborhood of the interface (localization), but it also tries to put as many monomers as possible in the oil (delocalization). We will study a new localization strategy, which consists in targeting the sites where the polymer touches the interface. In that way we explain why the polymer can be localized with prices of negative expectation.

Krishnamurthi RAVISHANKAR (SUNY - New Paltz)

Convergence of the coalescing random walk lattice filling curve to the Tóth-Werner plane filling curve.

In this joint work with C.M. Newman, we consider the lattice filling curve described by Tóth and Werner which forms the boundary between forward and backward coalescing random walks starting from even and odd space-time sub-lattices of Z² respectively. We show that this lattice filling curve converges in the diffusive scaling limit to a plane filling curve which is the boundary between the forward and backward Brownian webs. A one-dimensional projection of the two-dimensional result proves the convergence of Tóth's self-repelling walk to the Tóth-Werner continuum self repelling motion. The main new result is the tightness of the rescaled lattice model distributions.