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

Consider a branching Brownian motion (BBM). It is well known \cite{Bramson1983ConvergenceOS, Lalley1987ACL} that the rightmost particle is located near $m_t=\sqrt{2}t-\frac{3}{2\sqrt{2}}\log t$. Let $\mathcal{N}(t,x)$ be the set of particles within distance $x$ from $m_t$, where $x=o(t)$ grows with $t$. We prove that $\mathrm{\#}\mathcal{N}(t,x)\mathrm{/}{\pi }^{-1\mathrm{/}2}x{e}^{x{m}_t\mathrm{/}t}{e}^{-x^2\mathrm{/}(2t)}$ converges in probability to $Z_{\mathrm{\infty }}$, the limit of the so-called derivative martingale, and that, for $x=O(t^{1\mathrm{/}3})$, the convergence cannot be strengthened to an almost sure result. Moreover, we prove that the asymptotic overlap distribution of two particles sampled uniformly from $\mathcal{N}(t,x)$ converges to that of the critical derivative martingale measure. This establishes a universal genealogical picture of the BBM front at sublinear distances from the tip.