Simple exclusion process and last passage percolation

January 25, 2005, IHP, Amphi Darboux

Moderator: Pablo A. Ferrari (São Paulo & IHES)

Omer Angel (Vancouver): The stationary measure for TASEP with second class particles

We give a description of the stationary measure for a totally asymmetric exclusion process (TASEP) with second class particles, on either $\Z$ or on the cycle $\Z_N$. The measure is the image by a simple function of the uniform measure on a larger finite state space. This reveals a combinatorial mechanism behind several results on the TASEP with second class particles. We also discuss possible extensions to more classes of particles.

Bernard Derrida (ENS, Paris): Fluctuations and large deviations in the ASEP with open boundaries

This talk will present a short review on recent results obtained for the Asymmetric exclusion process with open boundaries: the expression and the time evolution of the large deviation of the density profile and the expression of the density fluctuations as a sum of a Brownian process and of an excursion process.

Pablo A. Ferrari (São Paulo & IHES): Second class particles, geodesics and competition interfaces

It is possible to linearly map a second class particle in the totally asymmetric simple exclusion process with a competition interface in the associated last passage percolation model. In the homogeneous case, the competition interface can be mapped to a geodesic. We discuss these relations and some of their consequences.

Patrik Ferrari (München): Stochastic growth in one dimension and Gaussian ensembles of random matrices

We consider a growth model in the Kardar-Parisi-Zhang universality class, the polynuclear growth (PNG) model, in one spatial dimension. For large growth time t, the scaling exponent of the fluctuations is 1/3 and the one of the spatial correlations is 2/3. The scaling functions of the fluctuations appear also in the Gaussian ensembles of random matrices (GRM). The connection between PNG and GRM is due to the same mathematical structure of some point processes associated to PNG and GRM. We discuss two geometries of the PNG, in which some determinantal / Pfaffian point processes show up, and compare with the corresponding GRM.

Joel L. Lebowitz (Rutgers & IHES): On the realizability of point processes with specified one and two particle densities

We investigate and give partial answers to the following question: given a one particle and pair density, $\rho_1({\bf r}_1)$ and $\rho_2({\bf r}_1,{\bf r}_2)$, ${\bf r}_1, {\bf r}_2 \in {\mathbb R}^d$ (or ${\mathbb Z}^d)$, does there exist a point process, i.e.\ a probability measure on points in ${\mathbb R}^d$ (${\mathbb Z}^d$), having these densities?
I will also discuss some results about superhomogeneous point processes.

James Martin (Paris): Last-passage percolation with general weight distribution

I'll give a review of a variety of results for last-passage percolation models with a general underlying distribution of the vertex weights. For example: conditions under which laws of large numbers for the passage times hold (giving shape theorems for the associated growth model); universality properties for paths close to one of the axes; entirely different scaling limits which occur when the tail of the distribution is sufficiently heavy (e.g. decaying polynomially with index less than 2).

Thomas Mountford (Lausanne): A Strong law of large numbers for the motion of a second class particle

(joint with H. Guiol) We consider the motion of a second class particle in a TASEP on the one dimensional integer lattice starting from heavyside initial data with positive shock. It is shown, as strongly suggested by preceding work of Ferrari and Kipnis, that a second class particle ``randomly picks" a speed among the interval of feasible candidates.

Gunter M. Schütz (Jülich): Broken ergodicity in driven one-dimensional reaction-diffusion systems with short-range interaction

Inspired by the motion of molecular motors along microtubuli we consider the one-dimensional totally asymmetric exclusion process with open boundaries, augmented by slow activated Langmuir kinetics which break the bulk particle number conservation. In the thermodynamic limit the steady state exhibits a phase with broken ergodicity and hysteresis which has no analog in systems investigated previously. Within a hydrodynamic approach we identify the main dynamical mode, viz. the random motion of a shock in an effective potential, which provides a general framework for understanding phase coexistence as well as ergodicity breaking in driven particle systems with slow reaction kinetics. This picture also leads to the exact phase diagram of such systems. (joint work with A. Rakos and M. Paessens).