Annals of Probability

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

Timo Seppäläinen

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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

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

First available in Project Euclid: 21 December 2001

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

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


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.

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