Asymptotics for the survival probability in a killed branching random walk. (English. French summary) Zbl 1210.60093

Summary: Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope \(\gamma - \varepsilon \), where \(\gamma \) denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when \(\varepsilon \rightarrow 0\), this probability decays like \(\exp\{ - \frac{\beta +o(1)}{\varepsilon ^{1/2}} \}\), where \(\beta \) is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli\((p)\) random variables (with \(0 < p < \frac12 \)) assigned on a rooted binary tree, this answers an open question of R. Pemantle [Ann. Appl. Probab. 19, No. 4, 1273–1291 (2009; Zbl 1176.68093)].


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)


Zbl 1176.68093
Full Text: DOI arXiv EuDML


[1] D. J. Aldous. A Metropolis-type optimization algorithm on the infinite tree. Algorithmica 22 (1998) 388-412. · Zbl 0936.68118 · doi:10.1007/PL00009231
[2] J. D. Biggins. The first- and last-birth problems for a multitype age-dependent branching process. Adv. in Appl. Probab. 8 (1976) 446-459. JSTOR: · Zbl 0339.60074 · doi:10.2307/1426138
[3] J. D. Biggins and A. E. Kyprianou. Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 (2005) 609-631, Paper 17. · Zbl 1110.60081
[4] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. · Zbl 0172.21201
[5] É. Brunet and B. Derrida. Shift in the velocity of a front due to a cutoff. Phys. Rev. E 56 (1997) 2597-2604. · doi:10.1103/PhysRevE.56.2597
[6] B. Derrida and D. Simon. The survival probability of a branching random walk in presence of an absorbing wall. Europhys. Lett. 78 (2007), Paper 60006. · Zbl 1244.82071 · doi:10.1209/0295-5075/78/60006
[7] B. Derrida and D. Simon. Quasi-stationary regime of a branching random walk in presence of an absorbing wall. J. Stat. Phys. 131 (2008) 203-233. · Zbl 1144.82321 · doi:10.1007/s10955-008-9504-4
[8] J. M. Hammersley. Postulates for subadditive processes. Ann. Probab. 2 (1974) 652-680. · Zbl 0303.60044 · doi:10.1214/aop/1176996611
[9] Y. Hu and Z. Shi. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009) 742-789. · Zbl 1169.60021 · doi:10.1214/08-AOP419
[10] K. Itô and H. P. McKean Jr. Diffusion Processes and Their Sample Paths . Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer, Berlin, 1974. · Zbl 0285.60063
[11] B. Jaffuel. The critical barrier for the survival of the branching random walk with absorption, 2009. Available at ArXiv math.PR/ .
[12] J.-P. Kahane and J. Peyrière. Sur certaines martingales de Mandelbrot. Adv. Math. 22 (1976) 131-145. · Zbl 0349.60051 · doi:10.1016/0001-8708(76)90151-1
[13] H. Kesten. Branching Brownian motion with absorption. Stochastic Process. Appl. 7 (1978) 9-47. · Zbl 0383.60077 · doi:10.1016/0304-4149(78)90035-2
[14] J. F. C. Kingman. The first birth problem for an age-dependent branching process. Ann. Probab. 3 (1975) 790-801. · Zbl 0325.60079 · doi:10.1214/aop/1176996266
[15] R. Lyons. A simple path to Biggins’ martingale convergence for branching random walk. In Classical and Modern Branching Processes 217-221. K. B. Athreya and P. Jagers (Eds). IMA Volumes in Mathematics and Its Applications 84 . Springer, New York, 1997. · Zbl 0897.60086
[16] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Probab. 23 (1995) 1125-1138. · Zbl 0840.60077 · doi:10.1214/aop/1176988176
[17] C. McDiarmid. Minimal positions in a branching random walk. Ann. Appl. Probab. 5 (1995) 128-139. · Zbl 0836.60089 · doi:10.1214/aoap/1177004832
[18] A. A. Mogulskii. Small deviations in the space of trajectories. Theory Probab. Appl. 19 (1974) 726-736. · Zbl 0326.60061
[19] R. Pemantle. Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009) 1273-1291. · Zbl 1176.68093 · doi:10.1214/08-AAP585
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.