Ellen Saada

Microscopic structure at the shock in the asymmetric simple exclusion

We study for a semi-infinite one dimensional initial distribution the asymptotic behaviour in the hydrodynamical limit at the shock. In this case the location of the shock is naturally identified by the position of the leftmost particle of the system for which we prove a central limit theorem. From this we deduce that at the shock local equilibrium does not hold

Euler hydrodynamics of one-dimensional attractive particle systems

We consider attractive irreducible conservative particle systems on Z, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1-30], from which we relax the assumption that the process has explicit invariant measures.

Couplings, attractiveness and hydrodynamics for conservative particle systems

Attractiveness is a fundamental tool to study interacting particle systems and the basic coupling construction is a usual route to prove this property, as for instance in simple exclusion. The derived Markovian coupled process (�t,�t)t�0 satisfies: (A) if �0 ≤ �0 (coordinate-wise), then for all t ≥ 0, �t ≤ �t a.s. In this paper, we consider generalized misanthrope models which are conservative par- ticle systems on Z d such that, in each transition, k particles may jump from a site x to another site y, with k ≥ 1. These models include simple exclusion for which k = 1, but, beyond that value, the basic coupling construction is not possible and a more refined one is required. We give necessary and sufficient conditions on the r to insure attractiveness; we construct a Markovian coupled process which both satisfies (A) and makes discrepan- cies between its two marginals non-increasing. We determine the extremal invariant and translation invariant probability measures under general irreducibility conditions. We ap- ply our results to examples including a two-species asymmetric exclusion process with charge conservation (for which k ≤ 2) which arises from a Solid-on-Solid interface dynam- ics, and a stick process (for which k is unbounded) in correspondence with a generalized discrete Hammersley-Aldous-Diaconis model. We derive the hydrodynamic limit of these two one-dimensional models.

Approaching criticality via the zero dissipation limit in the abelian avalanche model

The discrete height abelian sandpile model was introduced by Bak, Tang, Wiesenfeld and Dhar as an example for the concept of self-organized criticality. When the model is modified to allow grains to disappear on each toppling, it is called bulk-dissipative. We provide a detailed study of a continuous height version of the abelian sandpile model, called the abelian avalanche model, which allows an arbitrarily small amount of dissipation to take place on every toppling. We prove that for non-zero dissipation, the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure. We show that in the whole non-zero dissipation regime, the model is not critical, i.e., spatial covariances of local observables decay exponentially. We then study the zero dissipation limit and prove that the self-organized critical model is recovered, both for the stationary measure and for the dynamics. We obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality. We rigorously establish the mean-field value of this exponent for

Invariant measures of mass migration processes

d > 4

Asymmetric attractive particle systems on Z: Hydrodynamic limit for monotone initial profiles

d>4.

Macroscopic stability for nonfinite range kernels

We derive by a constructive method the hydrodynamic behavior of attractive processes with irreducible jumps and product invariant measures. Our approach relies on (i) explicit construction of Riemann solutions without assuming convexity, which may lead to contact discontinuities and (ii) a general result which proves that the hydrodynamic limit for Riemann initial profiles implies the same for general initial profiles. The k-step exclusion process provides a simple example. We also give a law of large numbers for the tagged particle in a nearest neighbor asymmetric k-step exclusion process.

Supercritical behavior of asymmetric zero-range process with sitewise disorder

We introduce the Mass Migration Process (MMP), a conservative particle system on