Exponential moments of truncated branching random walk martingales

Abstract: For a branching random walk that drifts to infinity, consider its Malthusian martingale, i.e.~the additive martingale with parameter $\theta$ being the smallest root of the characteristic equation. When particles are killed below the origin, we show that the limit of this martingale admits an exponential tail, contrary to the case without killing, where the tail is polynomial. In the critical case, where the characteristic equation has a single root, the same holds for the (truncated) derivative martingale, as we show. This study is motivated by recent work on first passage percolation on Erd\H{o}s-R\'enyi graphs.