×

Brunet-Derrida behavior of branching-selection particle systems on the line. (English) Zbl 1247.60124

The authors consider a class of branching-selection particle systems on \(R\) similar to the one considered by E. Brunet and B. Derrida [Computer Phys. Commun. 121–122, 376–381 (1999)] and give a rigorous mathematical proof 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 \( (\log N)^{-2}\).

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Athreya, K.B., Ney, P.E.: Branching processes. Mineola, NY: Dover Publications Inc., 2004. Reprint of original, New York: Springer, 1972 · Zbl 0259.60002
[2] Benguria R., Depassier M.C.: On the speed of pulled fronts with a cutoff. Phys. Rev. E 75, 051106 (2007) · doi:10.1103/PhysRevE.75.051106
[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) · doi:10.1140/epjb/e2008-00069-1
[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) · doi:10.1103/PhysRevE.76.041104
[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) · doi:10.1103/PhysRevE.56.2597
[7] Brunet É., Derrida B.: Microscopic models of traveling wave equations. Computer Phys. Commun. 121-122, 376–381 (1999) · doi:10.1016/S0010-4655(99)00358-6
[8] Brunet É., Derrida B.: Effect of microscopic noise on front propagation. J. Stat. Phys. 103(1-2), 269–282 (2001) · Zbl 1018.82020 · doi:10.1023/A:1004875804376
[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) · Zbl 1095.82008 · doi:10.1007/s10955-005-5960-2
[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) · Zbl 1244.82071
[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) · Zbl 1139.35056 · doi:10.1088/0951-7715/20/4/004
[12] Durrett, R.: Probability: theory and examples. Belmont, CA: Duxbury Press, second edition, 1996 · Zbl 1202.60001
[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 · Zbl 1210.60093
[14] Mueller C., Mytnik L., Quastel J.: Small noise asymptotics of traveling waves. Markov Process. Related Fields 14(3), 333–342 (2008) · Zbl 1155.60325
[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 · Zbl 1222.35105
[16] Pemantle R.: Search cost for a nearly optimal path in a binary tree. Ann. Appl. Prob. 19(4), 1273–1291 (2009) · Zbl 1176.68093 · doi:10.1214/08-AAP585
[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) · Zbl 1144.82321 · doi:10.1007/s10955-008-9504-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.