Maximal displacement of branching brownian motion

It is shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 21/2t–3 · 2−3/2 log t + O(1) at time t, the second-order term having been previously unknown. This determines (to within O(1)) the position of the travelling wave of the semilinear heat equation, ut =1/2uxx +f(u), in the classic paper by Kolmogorov-Petrovsky-Piscounov, “Etude de l'equations de la diffusion avec croissance de la quantite de la matiere et son application a un probleme biologique”, 1937.