One-dimensional random walks with self-blocking immigration

@article{Birkner2014OnedimensionalRW,
  title={One-dimensional random walks with self-blocking immigration},
  author={Matthias C. F. Birkner and Rongfeng Sun},
  journal={arXiv: Probability},
  year={2014}
}
We consider a system of independent one-dimensional random walkers where new particles are added at the origin at fixed rate whenever there is no older particle present at the origin. A Poisson ansatz leads to a semi-linear lattice heat equation and predicts that starting from the empty configuration the total number of particles grows as $c \sqrt{t} \log t$. We confirm this prediction and also describe the asymptotic macroscopic profile of the particle configuration. 
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