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

Abstract

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 \(\#\mathcal{N}(t,x)/\pi^{-1/2}xe^{xm_t/t} e^{-x^2/(2t)} \) converges in probability to $Z_\infty$, the limit of the so-called derivative martingale, and that, for \( x = O( t^{1/3}) \), 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.