## Annals of Probability

- Ann. Probab.
- Volume 29, Number 1 (2001), 176-204.

### Perturbation of the equilibrium for a totally asymmetric stick process in one dimension

#### Abstract

We study the evolution of a small perturbation of the equilibrium of a totally asymmetric one-dimensional interacting system. The model we take as an example is Hammersley's process as seen from a tagged particle, which can be viewed as a process of interacting positive-valued stick heights on the sites of $\mathbf{Z}$. It is known that under Euler scaling (space and time scale $n$) the empirical stick profile obeys the Burgers equation. We refine this result in two ways. If the process starts close enough to equilibrium, then over times $n^\nu$ for $1 \le \nu < 3$, and up to errors that vanish in hydrodynamic scale, the dynamics merely translates the initial stick configuration. In particular, on the hydrodynamic time scale, diffusive fluctuations are translated rigidly. A time evolution for the perturbation is visible under a particular family of scalings:over times $n_{\nu}, 1 < \nu < 3/2$, a perturbation of order $n^{1-\nu}$ from equilibrium follows the inviscid Burgers equation. The results for the stick model are derived from asymptotic results for tagged particles in Hammersley's process.

#### Article information

**Source**

Ann. Probab., Volume 29, Number 1 (2001), 176-204.

**Dates**

First available in Project Euclid: 21 December 2001

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1008956327

**Digital Object Identifier**

doi:10.1214/aop/1008956327

**Mathematical Reviews number (MathSciNet)**

MR1825147

**Zentralblatt MATH identifier**

1014.60091

**Subjects**

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

Secondary: 82C22: Interacting particle systems [See also 60K35]

**Keywords**

Perturbation of equilibrium hydrodynamic limit Hammersley’s process increasing sequences tagged particle

#### Citation

Seppäläinen, Timo. Perturbation of the equilibrium for a totally asymmetric stick process in one dimension. Ann. Probab. 29 (2001), no. 1, 176--204. doi:10.1214/aop/1008956327. https://projecteuclid.org/euclid.aop/1008956327