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, “Étude de l'équations de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique”, 1937.