The Annals of Applied Probability

Poissonian statistics in the extremal process of branching Brownian motion

Louis-Pierre Arguin, Anton Bovier, and Nicola Kistler

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Abstract

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647–1676] that, in the limit of large time $t$, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order $1$ from time $0$, or within a distance of order 1 from time $t$. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large $t$ to a random shift of a Poisson point process with exponential density. The last section discusses the Tidal Wave Conjecture by Lalley and Sellke [Ann. Probab. 15 (1987) 1052–1061] on the full limiting extremal process and its relation to the work of Chauvin and Rouault [Math. Nachr. 149 (1990) 41–59] on branching Brownian motion with atypical displacement.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 4 (2012), 1693-1711.

Dates
First available in Project Euclid: 10 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1344614208

Digital Object Identifier
doi:10.1214/11-AAP809

Mathematical Reviews number (MathSciNet)
MR2985174

Zentralblatt MATH identifier
1255.60152

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G70: Extreme value theory; extremal processes 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Branching Brownian motion extreme value theory extremal process traveling waves

Citation

Arguin, Louis-Pierre; Bovier, Anton; Kistler, Nicola. Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22 (2012), no. 4, 1693--1711. doi:10.1214/11-AAP809. https://projecteuclid.org/euclid.aoap/1344614208


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References

  • [1] Aidekon, E., Berestycki, J., Brunet, E. and Shi, Z. (2011). The branching Brownian motion seen from its tip. Available at arXiv:1104.3738.
  • [2] Arguin, L. P., Bovier, A. and Kistler, N. (2011). The extremal process of branching Brownian motion. Available at arXiv:1103.2322.
  • [3] Arguin, L. P., Bovier, A. and Kistler, N. (2011). Genealogy of extremal particles of branching Brownian motion. Comm. Pure Appl. Math. 64 1647–1676.
  • [4] Aronson, D. G. and Weinberger, H. F. (1975). Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974). Lecture Notes in Math. 446 5–49. Springer, Berlin.
  • [5] Aronson, D. G. and Weinberger, H. F. (1978). Multidimensional nonlinear diffusion arising in population genetics. Adv. in Math. 30 33–76.
  • [6] Bovier, A. and Kurkova, I. (2004). Derrida’s generalized random energy models. II. Models with continuous hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40 481–495.
  • [7] Bramson, M. (1983). Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44.
  • [8] Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531–581.
  • [9] Brunet, E. (2010). Private communication.
  • [10] Brunet, E. and Derrida, B. (2009). Statistics at the tip of a branching random walk and the delay of traveling waves. Eurphys. Lett. 87 60010.
  • [11] Brunet, É. and Derrida, B. (2011). A branching random walk seen from the tip. J. Stat. Phys. 143 420–446.
  • [12] Carpentier, D. and Le Doussal, P. (2001). Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon models. Phys. Rev. E 63 026110.
  • [13] Chauvin, B. and Rouault, A. (1990). Supercritical branching Brownian motion and K-P-P equation in the critical speed-area. Math. Nachr. 149 41–59.
  • [14] Derrida, B. and Spohn, H. (1988). Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51 817–840.
  • [15] Fisher, R. A. (1937). The wave of advance of advantageous genes. Ann. Eugen. 7 355–369.
  • [16] Fyodorov, Y. V. and Bouchaud, J.-P. (2008). Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 372001.
  • [17] Harris, S. C. (1999). Travelling-waves for the FKPP equation via probabilistic arguments. Proc. Roy. Soc. Edinburgh Sect. A 129 503–517.
  • [18] Kabluchko, Z. (2010). Private communication.
  • [19] Kessler, D. A., Levine, H., Ridgway, D. and Tsimring, L. (1997). Evolution on a smooth landscape. J. Stat. Phys. 87 519–544.
  • [20] Kolmogorov, A., Petrovsky, I. and Piscounov, N. (1937). Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscou Universitet, Bull. Math. 1 1–25.
  • [21] Lalley, S. P. and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 1052–1061.
  • [22] Maillard, P. (2010). A characterisation of superposable random measures. Available at arXiv:1102.1888.
  • [23] McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28 323–331.
  • [24] Munier, S. and Peschanski, R. (2004). Traveling wave fronts and the transition to saturation. Phys. Rev. D 69 034008.