The distance between the two BBM leaders

2020/10/20 by Berestycki, Julien, Brunet, Éric, Graham, Cole +3
#35B40 #35K57 #60J80 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2010.10431

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 d₁₂ 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(d₁₂ > a) as a→+∞. These tail asymptotics were previously known to "exponential order;" we discover an algebraic correction to this behavior.

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