Access by Xinjiang University

Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion

Bernard Derrida^*

Baruch Meerson^†

Pavel V. Sasorov^‡

Collège de France, 11 Place Marcelin Berthelot, 75005 Paris, France and Laboratoire de Physique Statistique, École Normale Supérieure, 24 Rue Lhomond, 75005 Paris, France

Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

Keldysh Institute of Applied Mathematics, Moscow 125047, Russia

^*derrida@lps.ens.fr
^†meerson@mail.huji.ac.il
^‡pavel.sasorov@gmail.com

PDF

Phys. Rev. E 93, 042139 – Published 29 April, 2016

DOI: https://doi.org/10.1103/PhysRevE.93.042139

Abstract

Consider a one-dimensional branching Brownian motion and rescale the coordinate and time so that the rates of branching and diffusion are both equal to 1. If $X_{1} (t)$ is the position of the rightmost particle of the branching Brownian motion at time $t$ , the empirical velocity $c$ of this rightmost particle is defined as $c = X_{1} (t) / t$ . Using the Fisher-Kolmogorov-Petrovsky-Piscounov equation, we evaluate the probability distribution $P (c, t)$ of this empirical velocity $c$ in the long-time $t$ limit for $c > 2$ . It is already known that, for a single seed particle, $P (c, t) \sim exp [- (c^{2} / 4 - 1) t]$ up to a prefactor that can depend on $c$ and $t$ . Here we show how to determine this prefactor. The result can be easily generalized to the case of multiple seed particles and to branching random walks associated with other traveling-wave equations.

Physics Subject Headings (PhySH)

Article Text

References (22)

H. P. McKean, Commun. Pure Appl. Math. 28, 323 (1975).
M. D. Bramson, Mem. Am. Math. Soc. 44 (285) (1983).
É. Brunet and B. Derrida, Europhys. Lett. 87, 60010 (2009); J. Stat. Phys. 143, 420 (2011).
A. H. Mueller and S. Munier, Phys. Rev. E 90, 042143 (2014).
K. Ramola, S. N. Majumdar, and G. Schehr, Phys. Rev. Lett. 112, 210602 (2014); Chaos, Solitons & Fractals 74, 79 (2015); Phys. Rev. E 91, 042131 (2015).
R. A. Fisher, Ann. Eugenics 7, 355 (1937).
A. Kolmogorov, I. Petrovsky, and N. Piscounov, Bull. Univ. État Moscou A 1, 1 (1937).
J. D. Murray, Mathematical Biology. I: An Introduction (Springer, New York, 2003).
J. F. Douglas, K. Efimenko, D. A. Fischer, F. R. Phelan, and J. Genzer, Proc. Natl. Acad. Sci. USA 104, 10324 (2007).
B. Derrida and H. Spohn, J. Stat. Phys. 51, 817 (1988); D. Carpentier and P. Le Doussal, Nucl. Phys. B 588, 565 (2000).
S. N. Majumdar and P. L. Krapivsky, Physica A 318, 161 (2003).
S. Munier and R. Peschanski, Phys. Rev. Lett. 91, 232001 (2003); I. Balitsky, Nucl. Phys. B 463, 99 (1996); Y. V. Kovchegov, Phys. Rev. D 60, 034008 (1999); 61, 074018 (2000); C. Marquet, R. Peschanski, and G. Soyez, Nucl. Phys. A 756, 399 (2005).
W. van Saarloos, Phys. Rep. 386, 29 (2003).
É. Brunet and B. Derrida, Phys. Rev. E 56, 2597 (1997).
U. Ebert and W. van Saarloos, Physica D 146, 1 (2000).
B. Chauvin and A. Rouault, Probab. Theory Relat. Fields 80, 299 (1988); A. Rouault, Pliska Stud. Math. Bulgar. 13, 15 (2000).
The asymptote (12) corresponds to the case of $β = 3 / 2$ in Eqs. (A2), (A6), (A8), and (A13) of Ref. [14]. It is also presented in Eq. (1.14) of Ref. [15].
This asymptote for $g_{1} (z \to \infty)$ also follows from the (quite cumbersome) exact solution of Eq. (25).
B. M. Budak and S. V. Fomin, Multiple Integrals, Field Theory and Series: An Advanced Course in Higher Mathematics (Mir, Moscow, 1978), p. 444, Theorem 10.9.
A. P. Prudnikov, I. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions (CRC, London, 1998).
See B. M. Budak and S. V. Fomin, Multiple Integrals, Field Theory and Series: An Advanced Course in Higher Mathematics (Ref. [19]), p. 443, Theorem 10.8.
Here we use the asymptote $u (x, t) ≃ α t^{- 3 / 2} (x - 2 t) exp (t - x^{2} / 4 t)$ only in the region $ln t ≪ x - 2 t \sim \sqrt{t}$ , which dominates the integral $J_{\infty}$ . As we will show shortly, the applicability domain of this asymptote is actually much broader: $ln t ≪ x - 2 t ≪ t$ .

Physical Review E

Export citation

Export citation

Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion

Abstract

Physics Subject Headings (PhySH)

Article Text

References (22)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation