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.
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References
Athreya, K.B., Ney, P.E.: Branching processes. Mineola, NY: Dover Publications Inc., 2004. Reprint of original, New York: Springer, 1972
Benguria R., Depassier M.C.: On the speed of pulled fronts with a cutoff. Phys. Rev. E 75, 051106 (2007)
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)
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
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)
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)
Brunet É., Derrida B.: Microscopic models of traveling wave equations. Computer Phys. Commun. 121-122, 376–381 (1999)
Brunet É., Derrida B.: Effect of microscopic noise on front propagation. J. Stat. Phys. 103(1-2), 269–282 (2001)
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)
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)
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)
Durrett, R.: Probability: theory and examples. Belmont, CA: Duxbury Press, second edition, 1996
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
Mueller C., Mytnik L., Quastel J.: Small noise asymptotics of traveling waves. Markov Process. Related Fields 14(3), 333–342 (2008)
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
Pemantle R.: Search cost for a nearly optimal path in a binary tree. Ann. Appl. Prob. 19(4), 1273–1291 (2009)
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)
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Communicated by H. Spohn
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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
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DOI: https://doi.org/10.1007/s00220-010-1067-y