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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.01893 |
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| _version_ | 1866929556121714688 |
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| author | Redig, Frank Saada, Ellen |
| author_facet | Redig, Frank Saada, Ellen |
| contents | 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 $\rightarrow$ $\infty$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Particle systems with sources and sinks Redig, Frank Saada, Ellen Probability 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 $\rightarrow$ $\infty$ 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. |
| title | Particle systems with sources and sinks |
| topic | Probability |
| url | https://arxiv.org/abs/2403.01893 |