The Annals of Applied Probability

One-dimensional random walks with self-blocking immigration

Matthias Birkner and Rongfeng Sun

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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

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.


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References

  • [1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. U.S. Government Printing Office, Washington, DC.
  • [2] Birkner, M. (2003). Particle systems with locally dependent branching: Long-time behaviour, genealogy and critical parameters. Dissertation, Johann Wolfgang Goethe-Universität Frankfurt am Main.
  • [3] Birkner, M. and Sun, R. (2016). Low-dimensional lonely branching random walks die out. Manuscript in progress.
  • [4] Chen, Z.-Q. and Fan, W.-T. (2016). Hydrodynamic limits and propagation of chaos for interacting random walks in domains. Ann. Appl. Probab. To appear. Available at arXiv:1311.2325.
  • [5] Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
  • [6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [7] Garcia, N. L. and Kurtz, T. G. (2006). Spatial birth and death processes as solutions of stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat. 1 281–303.
  • [8] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 320. Springer, Berlin.
  • [9] Newman, C. M., Ravishankar, K. and Sun, R. (2005). Convergence of coalescing nonsimple random walks to the Brownian web. Electron. J. Probab. 10 21–60.
  • [10] Yau, H.-T. (1991). Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22 63–80.