2024/03/04 by Frank Redig, Redig, Frank, Ellen Saada +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2403.01893
openalex publication_date 2024/03/04 · openalex created_date 2024/03/06 · openalex updated_date 2026/08/01
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.