## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 2 (1994), 820-832.

### Current Fluctuations for the Asymmetric Simple Exclusion Process

P. A. Ferrari and L. R. G. Fontes

#### Abstract

We compute the diffusion coefficient of the current of particles through a fixed point in the one-dimensional nearest neighbor asymmetric simple exclusion process in equilibrium. We find $D = |p - q|\rho(1 - \rho)|1 - 2\rho|$, where $p$ is the rate at which the particles jump to the right, $q$ is the jump rate to the left and $\rho$ is the density of particles. Notice that $D$ vanishes if $p = q$ or $\rho = 1/2$. Laws of large numbers and central limit theorems are also proven. Analogous results are obtained for the current of particles through a position travelling at a deterministic velocity $r$. As a corollary we get that the equilibrium density fluctuations at time $t$ are a translation of the fluctuations at time 0. We also show that the current fluctuations at time $t$ are given, in the scale $t^{1/2}$, by the initial density of particles in an interval of length $|(p - q)(1 - 2\rho)|t$. The process is isomorphic to a growth interface process. Our result means that the equilibrium growth fluctuations depend on the general inclination of the surface. In particular, they vanish for interfaces roughly perpendicular to the observed growth direction.

#### Article information

**Source**

Ann. Probab. Volume 22, Number 2 (1994), 820-832.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988731

**Digital Object Identifier**

doi:10.1214/aop/1176988731

**Mathematical Reviews number (MathSciNet)**

MR1288133

**Zentralblatt MATH identifier**

0806.60099

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82C22: Interacting particle systems [See also 60K35] 82C24: Interface problems; diffusion-limited aggregation 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

**Keywords**

Asymmetric simple exclusion current fluctuations driven interface

#### Citation

Ferrari, P. A.; Fontes, L. R. G. Current Fluctuations for the Asymmetric Simple Exclusion Process. Ann. Probab. 22 (1994), no. 2, 820--832. doi:10.1214/aop/1176988731. https://projecteuclid.org/euclid.aop/1176988731.