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

TL;DR

This work advances the understanding of the BBM front at sublinear distances by quantifying the number of particles within $m_t-x$ when $x=o(t/\log t)$ and showing $N(t,x)/f(t,x)$ converges in probability to the derivative martingale limit $Z_\infty$, with $f(t,x)=\pi^{-1/2}x e^{\sqrt{2}x} e^{-x^2/(2t)}$. The authors establish a path-localisation picture: particles contributing to $N(t,x)$ mostly follow trajectories largely confined below a barrier and branch relatively early, and they connect these local genealogies to the SDPPP description of the extremal process. They also prove that, for $x_t\le t^{1/3}$, convergence cannot be improved to almost sure, highlighting intrinsic fluctuations near the front. Techniques combine barrier arguments, Brownian bridge estimates, and many-to-one/two lemmas to derive both first- and second-mole bounds, yielding a coherent view of the BBM front and its relation to existing extremal-process results and Gibbs-measure perspectives. Overall, the results deepen the link between BBM extremal statistics, path localisation, and log-correlated field universality, with implications for related branching processes and reaction-diffusion models.

Abstract

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)/π^{-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 $.

Figures (1)

• Figure 1: Illustration of a rare event—which occurs infinitely often almost surely—where $M_t$ exceeds $m_t + \frac{\log(t)}{\sqrt{2}}$, resulting in an inflation of $N(s, x_s)$ for $x_s \leq s^{1/3}$ up to time $t + t^{2/3}$.

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Conjecture 1.6
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.4