The distance between the two BBM leaders

Julien Berestycki (1) , Éric Brunet (2) , Cole Graham (3) , Leonid Mytnik (4) , Jean-Michel Roquejoffre (5) , Lenya Ryzhik (3)

1 University of Oxford
2 Systèmes Classiques ou Quantiques en Interaction
3 Stanford University
4 Technion - Israel Institute of Technology [Haifa]
5 IMT - Institut de Mathématiques de Toulouse UMR5219

Abstract We study the distance between the two rightmost particles in branching Brownian motion. Derrida and the second author have shown that the long-time limit d 12 of this random variable can be expressed in terms of PDEs related to the Fisher–KPP equation. We use such a representation to determine the sharp asymptotics of P ( d 12 > a ) as a → +∞. These tail asymptotics were previously known to ‘exponential order;’ we discover an algebraic correction to this behavior.

Keywords: branching Brownian motion; Fisher–KPP equation; branching processes

Domains: Mathematics [math]; Analysis of PDEs [math.AP]

Julien Berestycki, Éric Brunet, Cole Graham, Leonid Mytnik, Jean-Michel Roquejoffre, et al.. The distance between the two BBM leaders. Nonlinearity, 2022, 35 (4), pp.1558-1609. ⟨10.1088/1361-6544/ac4a8e⟩. ⟨hal-03669216⟩