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

Abstract

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}.