Skip to main content
Log in

Brunet-Derrida Behavior of Branching-Selection Particle Systems on the Line

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider a class of branching-selection particle systems on \({\mathbb{R}}\) similar to the one considered by E. Brunet and B. Derrida in their 1997 paper “Shift in the velocity of a front due to a cutoff”. Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size N of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate (log N)−2. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of N independent branching random walks killed below a linear space-time barrier.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Athreya, K.B., Ney, P.E.: Branching processes. Mineola, NY: Dover Publications Inc., 2004. Reprint of original, New York: Springer, 1972

  2. Benguria R., Depassier M.C.: On the speed of pulled fronts with a cutoff. Phys. Rev. E 75, 051106 (2007)

    Article  ADS  Google Scholar 

  3. Benguria R., Depassier M.C., Loss M.: Validity of the Brunet-Derrida formula for the speed of pulled fronts with a cutoff. Eur. Phys. J B 61, 331 (2008)

    Article  ADS  Google Scholar 

  4. Bérard, J.: An example of Brunet-Derrida behavior for a branching-selection particle system on Z. http://arxiv.org/abs/0810.5567v3[math.PR], 2008

  5. Brunet É., Derrida B., Mueller A.H., Munier S.: Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76(4), 041104 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  6. Brunet E., Derrida B.: Shift in the velocity of a front due to a cutoff. Phys. Rev. E (3) 56(3, part A), 2597–2604 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  7. Brunet É., Derrida B.: Microscopic models of traveling wave equations. Computer Phys. Commun. 121-122, 376–381 (1999)

    Article  ADS  Google Scholar 

  8. Brunet É., Derrida B.: Effect of microscopic noise on front propagation. J. Stat. Phys. 103(1-2), 269–282 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Conlon J.G., Doering C.R.: On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky- Piscunov equation. J. Stat. Phys. 120(3-4), 421–477 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Derrida, B., Simon, D.: The survival probability of a branching random walk in presence of an absorbing wall. Europhys. Lett. EPL 78(6), Art. 60006, 6 (2007)

    Google Scholar 

  11. Dumortier F., Popović N., Kaper T.J.: The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off. Nonlinearity 20(4), 855–877 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Durrett, R.: Probability: theory and examples. Belmont, CA: Duxbury Press, second edition, 1996

    Google Scholar 

  13. Gantert, N., Hu, Y., Shi, Z.: Asymptotics for the survival probability in a supercritical branching random walk. http://arxiv.org/abs/0811.0262v2[math.PR], 2008

  14. Mueller C., Mytnik L., Quastel J.: Small noise asymptotics of traveling waves. Markov Process. Related Fields 14(3), 333–342 (2008)

    MATH  MathSciNet  Google Scholar 

  15. Mueller, C., Mytnik, L., Quastel, J.: Effect of noise on front propagation in reaction-diffusion equations of KPP type. http://arxiv.org/abs/0902.3423v1[math.PR], 2009

  16. Pemantle R.: Search cost for a nearly optimal path in a binary tree. Ann. Appl. Prob. 19(4), 1273–1291 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Simon D., Derrida B.: Quasi-stationary regime of a branching random walk in presence of an absorbing wall. J. Stat. Phys. 131(2), 203–233 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jean-Baptiste Gouéré.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bérard, J., Gouéré, JB. Brunet-Derrida Behavior of Branching-Selection Particle Systems on the Line. Commun. Math. Phys. 298, 323–342 (2010). https://doi.org/10.1007/s00220-010-1067-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-010-1067-y

Keywords

Navigation