Electronic Journal of Probability

Dependent double branching annihilating random walk

Marton Balazs and Attila Nagy

Full-text: Open access

Abstract

Double (or parity conserving) branching annihilating random walk, introduced by Sudbury in '90, is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved by Belitsky, Ferrari, Menshikov and Popov in '01 and, subsequently in a much more general setup, in the article by Sturm and Swart (Tightness of voter model interfaces) in '08. These results assume that jump rates of the various moves do not depend on the configuration of the particles not involved in these moves. The present article deals with the case when the jump rates are affected by the locations of several particles in the system. Motivation for such models comes from non-attractive interacting particle systems with particle conservation. Under suitable assumptions we establish the existence of the process, and prove that the one-particle state is positive recurrent. We achieve this by arguments similar to those appeared in the previous article by Sturm and Swart. We also extend our results to some cases of long range jumps, when branching can also occur to non-neighboring sites. We outline and discuss several particular examples of models where our results apply.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 84, 32 pp.

Dates
Accepted: 13 August 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067190

Digital Object Identifier
doi:10.1214/EJP.v20-4045

Mathematical Reviews number (MathSciNet)
MR3383568

Zentralblatt MATH identifier
1329.60327

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
non-attractive particle system long range dependent rates double branching annihilating random walk parity conserving positive recurrence interface tightness

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Balazs, Marton; Nagy, Attila. Dependent double branching annihilating random walk. Electron. J. Probab. 20 (2015), paper no. 84, 32 pp. doi:10.1214/EJP.v20-4045. https://projecteuclid.org/euclid.ejp/1465067190


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References

  • Belhaouari, S.; Mountford, T.; Valle, G. Tightness for the interfaces of one-dimensional voter models. Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 421–442.
  • Belitsky, Vladimir; Ferrari, Pablo A.; Menshikov, Mikhail V.; Popov, Serguei Yu. A mixture of the exclusion process and the voter model. Bernoulli 7 (2001), no. 1, 119–144.
  • Blath, Jochen; Kurt, Noemi. Survival and extinction of caring double-branching annihilating random walk. Electron. Commun. Probab. 16 (2011), 271–282.
  • Bramson, M.; Calderoni, P.; De Masi, A.; Ferrari, P.; Lebowitz, J.; Schonmann, R. H. Microscopic selection principle for a diffusion-reaction equation. J. Statist. Phys. 45 (1986), no. 5-6, 905–920.
  • Bramson, Maury; Gray, Lawrence. The survival of branching annihilating random walk. Z. Wahrsch. Verw. Gebiete 68 (1985), no. 4, 447–460.
  • Bramson, Maury; Ding, Wan Ding; Durrett, Rick. Annihilating branching processes. Stochastic Process. Appl. 37 (1991), no. 1, 1–17.
  • Cardy, John L.; Täuber, Uwe C. Field theory of branching and annihilating random walks. J. Statist. Phys. 90 (1998), no. 1-2, 1–56.
  • Cardy, J. L. and Täber, U. C.: Theory of branching and annihilating random walk. phPhys. Rev. Lett. 77, (1996), 4780–4783. http://dx.doi.org/10.1103/PhysRevLett.77.4780
  • Cox, J. T.; Durrett, R. Hybrid zones and voter model interfaces. Bernoulli 1 (1995), no. 4, 343–370.
  • De Masi, A.; Ferrari, P. A.; Lebowitz, J. L. Reaction-diffusion equations for interacting particle systems. J. Statist. Phys. 44 (1986), no. 3-4, 589–644.
  • Griffeath, David. Additive and cancellative interacting particle systems. Lecture Notes in Mathematics, 724. Springer, Berlin, 1979. iv+108 pp. ISBN: 3-540-09508-X
  • Kipnis, Claude; Landim, Claudio. Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320. Springer-Verlag, Berlin, 1999. xvi+442 pp. ISBN: 3-540-64913-1
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4
  • Liggett, Thomas M. Stochastic interacting systems: contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer-Verlag, Berlin, 1999. xii+332 pp. ISBN: 3-540-65995-1
  • Odor, Geza. Universality in nonequilibrium lattice systems. Theoretical foundations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. xx+276 pp. ISBN: 978-981-281-227-8; 981-281-227-X
  • Sturm, Anja; Swart, Jan M. Tightness of voter model interfaces. Electron. Commun. Probab. 13 (2008), 165–174.
  • Sturm, Anja; Swart, Jan. Voter models with heterozygosity selection. Ann. Appl. Probab. 18 (2008), no. 1, 59–99.
  • Sudbury, Aidan. Dual families of interacting particle systems on graphs. J. Theoret. Probab. 13 (2000), no. 3, 695–716.
  • Sudbury, Aidan. The branching annihilating process: an interacting particle system. Ann. Probab. 18 (1990), no. 2, 581–601.
  • Sudbury, Aidan. The survival of nonattractive interacting particle systems on $\mathbf{Z}$. Ann. Probab. 28 (2000), no. 3, 1149–1161.