L. R. G. Fontes

Shock fluctuations in the asymmetric simple exclusion process

SummaryWe consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the initial measure isvρ,λ, a product measure with densities ρ and λ to the left and right of the origin respectively, ρ<λ, then there exists a (microscopic) shock for the system. A shock is a random positionXt such that the system as seen from this position at timet has asymptotic product distributions with densities ρ and λ to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=limt→∞t−1(E(Xt)2−(EXt)2) and findD=(p−q)(λ−ρ)−1(ρ(1−ρ)+λ(1−λ)) as conjectured by Spohn (1991). We show that in the scale

Two-dimensional poisson trees converge to the Brownian web

\sqrt t

The Brownian Web

the position ofXt is determined by the initial distribution of particles in a region of length proportional tot. We prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities ρ and λ. This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.

POISSONIAN APPROXIMATION FOR THE TAGGED PARTICLE IN ASYMMETRIC SIMPLE EXCLUSION

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for p = pc+λδ1/ν, with ν = 4/3, as the lattice spacing δ → 0. Our proposed framework extends previous analyses for p = pc, based on SLE6. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of “macroscopically pivotal” lattice sites and the marked ones are those that actually change state as λ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.

K-processes, scaling limit and aging for the trap model in the complete graph

Abstract The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times in R and at all points of R . The two-dimensional Poisson tree is a family of continuous time one-dimensional random walks with uniform jumps in a bounded interval. The walks start at the space–time points of a homogeneous Poisson process in R 2 and are in fact constructed as a function of the point process. This tree was introduced by Ferrari, Landim and Thorisson. By verifying criteria derived by Fontes, Isopi, Newman and Ravishankar, we show that, when properly rescaled, and under the topology introduced by those authors, Poisson trees converge weakly to the Brownian web.

The asymmetric simple exclusion model with multiple shocks

We consider a process of two classes of particles jumping on a one-dimensional lattice. The marginal system of the first class of particles is the one-dimensional totally asymmetric simple exclusion process. When classes are disregarded the process is also the totally asymmetric simple exclusion process. The existence of a unique invariant measure with product marginals with density ρ and λ for the first- and first- plus second-class particles, respectively, was shown by Ferrari, Kipnis, and Saada. Recently Derrida, Janowsky, Lebowitz, and Speer have computed this invariant measure for finite boxes and performed the infinite-volume limit. Based on this computation we give a complete description of the measure and derive some of its properties. In particular we show that the invariant measure for the simple exclusion process as seen from a second-class particle with asymptotic densities ρ and λ is equivalent to the product measure with densities ρ to the left of the origin and λ to the right of the origin.

A Dynamic One-Dimensional Interface Interacting with a Wall

Arratia, [Arratia, R. (1979) Ph.D. thesis (University of Wisconsin, Madison) and unpublished work] and later Toth and Werner [Toth, B. & Werner, W. (1998) Probab. Theory Relat. Fields111, 375–452] constructed random processes that formally correspond to coalescing one-dimensional Brownian motions starting from every space-time point. We extend their work by constructing and characterizing what we call the Brownian Web as a random variable taking values in an appropriate (metric) space whose points are (compact) sets of paths. This leads to general convergence criteria and, in particular, to convergence in distribution of coalescing random walks in the scaling limit to the Brownian Web.

THE FULL BROWNIAN WEB AS SCALING LIMIT OF STOCHASTIC FLOWS

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor simple exclusion process. Each particle attempts to jump to the site to its right at rate p and to the site to its left at rate q. The jump is realized if the destination site is empty. We assume p > q. The initial distribution is the product measure with density