Scaling Limit of the Path Leading to the Leftmost Particle in a Branching Random Walk

Abstract

We consider a discrete-time branching random walk defined on the real line, which is assumed to be supercritical and in the boundary case. It is known that its leftmost position of the $n$th generation behaves asymptotically like $\frac{3}{2}\,\log n$, provided the nonextinction of the system. The main goal of this paper is to prove that the path from the root to the leftmost particle, after a suitable normalization, converges weakly to a Brownian excursion in $D([0,1],{\bf R})$.