arXiv subjects

Gabriel Flath

Publications and source records attributed to Gabriel Flath.

3 recordsLinked to original sources

arXiv2026

Almost sure path localisation for the derivative martingale of branching Brownian motion

The evolution of the front of branching Brownian motion is determined by the limit of the derivative martingale. In this work, we characterise which particles contribute to this limit. Precisely, we establish a sharp almost sure path localisation result which shows that the limit is determined by those particles whose trajectory stays within a thin tube at distance $s^{1/2}$ from the extremal particle.

math.PR↗

arXiv2025

A simpler path to Ergodic Theorems for the Frontier of Branching Brownian Motion

We revisit the ergodic theorem for the frontier of branching Brownian motion (BBM). Motivated by the proof of Arguin, Bovier, and Kistler \cite{arguin2012ergodic}, we provide a shorter and more direct argument. It relies on two observations: pairs of extremal particles observed at well-separated times must have branched early, and pairs of early-branching extremal particles have negatively correlated positions. This yields the ergodic theorem for BBM and extends it to a broad class of functionals of the recentred maximum. We also address a gap in the path localization argument of \cite{arguin2012ergodic}.

math.PR↗

arXiv2025

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 $\#\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.

math.PR↗