Particle systems with sources and sinks
arxiv(2024)
Abstract
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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