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Pré-Publication, Document De Travail Année : 2024

Particle systems with sources and sinks

Résumé

Local perturbations in conservative particle systems can have a non-local influence on the stationary measure. To capture this phenomenon, we analyze in this paper two toy models. We study the symmetric exclusion process on a countable set of sites V with a source at a given point (called the origin), starting from a Bernoulli product measure with density ρ. We prove that when the underlying random walk on V is recurrent, then the system evolves towards full occupation, whereas in the transient case we obtain a limiting distribution which is not product and has long-range correlations. For independent random walkers on V , we analyze the same problem, starting from a Poissonian measure. Via intertwining with a system of ODE's, we prove that the distribution is Poissonian at all later times t > 0, and that the system "explodes" in the limit t → ∞ if and only if the underlying random walk is recurrent. In the transient case, the limiting density is a simple function of the Green's function of the random walk.
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Dates et versions

hal-04487149 , version 1 (03-03-2024)

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  • HAL Id : hal-04487149 , version 1

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Frank Redig, Ellen Saada. Particle systems with sources and sinks. 2024. ⟨hal-04487149⟩
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