The distance between the two BBM leaders

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 PDE5 related to the Fisher-KPP equation. We use such a representation to determine the sharp asymptotics of P(d(12 )> a) as a -> +infinity. These tail asymptotics were previously known to 'exponential order;' we discover an algebraic correction to this behavior.