M. Balázs, M. Z. Rácz, and B. , Tóth: Modeling flocks and prices: jumping particles with an attractive interaction, 1107.

J. Baik, P. Deift, K. Percy, and . Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, Journal of the American Mathematical Society, vol.12, issue.04, pp.1119-1178, 1999.
DOI : 10.1090/S0894-0347-99-00307-0

J. Berestycki, N. Berestycki, and J. , Survival of Near-Critical Branching Brownian Motion, Journal of Statistical Physics, vol.131, issue.1, pp.833-854, 2011.
DOI : 10.1007/s10955-011-0224-9
URL : https://hal.archives-ouvertes.fr/hal-00661135

J. Bérard and J. Gouéré, Brunet-Derrida Behavior of Branching-Selection Particle Systems on the Line, Communications in Mathematical Physics, vol.131, issue.2, pp.323-342, 2010.
DOI : 10.1007/s00220-010-1067-y

J. Bérard and J. Gouéré, Survival Probability of the Branching Random Walk Killed Below a Linear Boundary, Electronic Journal of Probability, vol.16, issue.0, pp.396-418, 2011.
DOI : 10.1214/EJP.v16-861

J. Bertoin, Lévy processes. Cambridge Tracts in Math, 1996.

E. Brunet and B. Derrida, Exactly soluble noisy traveling-wave equation appearing in the problem of directed polymers in a random medium, Physical Review E, vol.70, issue.1, p.16106, 2004.
DOI : 10.1103/PhysRevE.70.016106
URL : https://hal.archives-ouvertes.fr/hal-00005373

E. Brunet, B. Derrida, A. Mueller, and S. Munier, Noisy traveling waves: Effect of selection on genealogies, Europhysics Letters (EPL), vol.76, issue.1, pp.1-7, 2006.
DOI : 10.1209/epl/i2006-10224-4

O. Catoni and R. Cerf, The exit path of a Markov chain with rare transitions, ESAIM: Probability and Statistics, vol.1, pp.95-144, 1995.
DOI : 10.1051/ps:1997105

S. Chatterjee and S. , A phase transition behavior for Brownian motions interacting through their ranks, Probability Theory and Related Fields, vol.75, issue.4, pp.123-159, 2010.
DOI : 10.1007/s00440-009-0203-0

O. Couronné and L. Gérin, Survival Time of a Censored Supercritical Galton-Watson Process. http://fr.arxiv.org/abs/1111.1078 , to appear in Ann, Inst. H. Poincaré ? Probabilités et Statistiques [12] A. Dembo, O. Zeitouni: Large deviations techniques and applications. Applications of Mathematics 38, 1998.

B. Derrida and H. , Polymers on disordered trees, spin glasses, and traveling waves, Journal of Statistical Physics, vol.46, issue.5-6, pp.817-840, 1988.
DOI : 10.1007/BF01014886

R. Durrett, Probability: theory and examples. Fourth edition, 2010.
DOI : 10.1017/CBO9780511779398

R. Durrett and D. , Remenik: Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations, 2011.

B. Gilding and R. , Kersner: Travelling waves in nonlinear diffusion-convection reaction, Progress in Nonlinear Differential Equations and their Applications, 2004.

I. Ibragimov and Y. Linnik, Independent and stationary sequences of random variables, 1971.

J. Jacod, Théorèmes limite pour les processus. Saint-Flour notes, Lecture Notes in Math, pp.298-409, 1117.

K. Johansson, Shape Fluctuations and Random Matrices, Communications in Mathematical Physics, vol.209, issue.2, pp.437-476, 2000.
DOI : 10.1007/s002200050027
URL : http://arxiv.org/abs/math/9903134

M. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and related properties of random sequences and processes, 1983.
DOI : 10.1007/978-1-4612-5449-2

P. Maillard, S. N. Majumdar, and P. Krapivsky, Branching Brownian motion with selection of the N right-most particles: An approximate model. http://arxiv.org/abs/1112.0266, to appear in Ann Extreme value statistics and traveling fronts: Application to computer science Phys, pp.65-036127, 2002.

C. Monthus and T. Garel, Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions, Journal of Physics A: Mathematical and Theoretical, vol.42, issue.7, p.75002, 2009.
DOI : 10.1088/1751-8113/42/7/075002

C. Mueller, L. Mytnik, and J. , Effect of noise on front propagation in??reaction-diffusion equations of KPP type, Inventiones mathematicae, vol.386, issue.1, pp.405-453, 2011.
DOI : 10.1007/s00222-010-0292-5

J. Nolen, A central limit theorem for pulled fronts in a random medium, Networks and Heterogeneous Media, vol.6, issue.2, pp.167-194, 2011.
DOI : 10.3934/nhm.2011.6.167

D. Panja, Effects of fluctuations on propagating fronts, Physics Reports, vol.393, issue.2, pp.87-174, 2004.
DOI : 10.1016/j.physrep.2003.12.001

S. Pal and J. Pitman, One-dimensional Brownian particle systems with rank-dependent drifts, The Annals of Applied Probability, vol.18, issue.6, pp.2179-2207, 2008.
DOI : 10.1214/08-AAP516

A. Ruzmaikina and M. Aizenman, Characterization of invariant measures at the leading edge for competing particle systems, The Annals of Probability, vol.33, issue.1, pp.82-113, 2005.
DOI : 10.1214/009117904000000865

G. Street, . Toronto, M. Ontario, and . 1l2, Canada E-mail address: quastel@math.toronto.edu (Alejandro F. Ramírez) Facultad de Matemáticas