The number of particles at sublinear distances from the tip in branching Brownian motion

Consider a branching Brownian motion (BBM). It is well known that the rightmost particle is located near $ m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t $. Let $N(t,x) $ be the number of particles within distance $ x $ from $ m_t $, where $ x = o(t/\log(t)) $ grows with $ t $. We prove that $ N(t,x)/\pi^{-1/2}xe^{\sqrt{2}x} e^{-x^2/(2t)} $ converges in probability to $Z_\infty$ and note that, for $ x \leq t^{1/3} $, the convergence cannot be strengthened to an almost sure result. Moreover, the intermediate steps in our proof provide a path localisation for the trajectories of particles in $ N(t,x) $ and their genealogy, allowing us to develop a picture of the BBM front at sublinear distances from $ m_t $.