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 distinct times far apart must have branched early, and pairs of early-branching extremal particles have negatively correlated positions. This yields the ergodic theorem for BBM. We also address a gap in the path localization argument of \cite{arguin2012ergodic}.