## The Annals of Applied Probability

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

### One-dimensional random walks with self-blocking immigration

Matthias Birkner and Rongfeng Sun

#### Abstract

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

**Source**

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

**Dates**

Received: October 2014

Revised: September 2015

First available in Project Euclid: 6 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1488790824

**Digital Object Identifier**

doi:10.1214/16-AAP1199

**Mathematical Reviews number (MathSciNet)**

MR3619784

**Zentralblatt MATH identifier**

1362.60082

**Subjects**

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

**Keywords**

Interacting random walks density-dependent immigration Poisson comparison vacant time

#### Citation

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. https://projecteuclid.org/euclid.aoap/1488790824