The Annals of Applied Probability

One-dimensional random walks with self-blocking immigration

Matthias Birkner and Rongfeng Sun

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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.

Article information

Ann. Appl. Probab., Volume 27, Number 1 (2017), 109-139.

Received: October 2014
Revised: September 2015
First available in Project Euclid: 6 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G50: Sums of independent random variables; random walks 60F99: None of the above, but in this section

Interacting random walks density-dependent immigration Poisson comparison vacant time


Birkner, Matthias; Sun, Rongfeng. One-dimensional random walks with self-blocking immigration. Ann. Appl. Probab. 27 (2017), no. 1, 109--139. doi:10.1214/16-AAP1199.

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