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Avena, Luca; Franco, Tertuliano; Jara, Milton; Völlering, Florian

Symmetric exclusion as a random environment: hydrodynamic limits. (English. French summary) Zbl 1359.82012

Ann. Inst. Henri Poincaré, Probab. Stat. 51, No. 3, 901-916 (2015).
The authors study random walks on the one-dimensional integer lattice \(\mathbb{Z}\), where the transition rates vary according to a simple symmetric exclusion process \(\eta_t\) on \(\mathbb{Z}\). More precisely, the random walk jumps to the left with rate \(\alpha\) and to the right with rate \(\beta\) if the same position in \(\eta_t\) is empty; if a particle is sitting on that position of \(\eta_t\), then the transition rates are reversed. This model creates trapping phenomena at the boundary between populated and empty regions of \(\eta_t\). Hence they are dealing with a random walk in a slowly non-uniform mixing dynamic random environment. As the main result, the authors prove a hydrodynamic limit theorem for the exclusion as seen by the walk. Moreover, a differential equation for the macroscopic evolution of the random walk limit is presented. For these purposes, time and space are scaled by \(N\) in time and by \(1/N\) in space, in contrast to the commonly used \(N^2\) and \(1/N\) scaling. Finally, some variants of this model are also treated.
Reviewer: Christoph Koutschan (Linz)

Cited in 8 Documents

MSC:

82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
60K37 Processes in random environments
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics

Keywords:

random walks in random environments; macroscopic speed; hydrodynamic limits; particle systems; exclusion process
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Full Text: DOI arXiv Euclid

References:

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