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February 2001 Perturbation of the equilibrium for a totally asymmetric stick process in one dimension
Timo Seppäläinen
Ann. Probab. 29(1): 176-204 (February 2001). DOI: 10.1214/aop/1008956327


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.


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Timo Seppäläinen. "Perturbation of the equilibrium for a totally asymmetric stick process in one dimension." Ann. Probab. 29 (1) 176 - 204, February 2001.


Published: February 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1014.60091
MathSciNet: MR1825147
Digital Object Identifier: 10.1214/aop/1008956327

Primary: 60K35
Secondary: 82C22

Keywords: Hammersley’s process , Hydrodynamic limit , increasing sequences , Perturbation of equilibrium , Tagged particle

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 1 • February 2001
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