Scaling limit of the path leading to the leftmost particle in a branching random walk

TL;DR

This paper proves that the normalized path to the leftmost particle in a supercritical branching random walk converges weakly to a Brownian excursion, revealing new insights into the path structure of such stochastic processes.

Contribution

It establishes the weak convergence of the path to the leftmost particle to a Brownian excursion, a novel result in the study of branching random walks.

Findings

Path converges to Brownian excursion after normalization

Leftmost position behaves asymptotically like (3/2)ln n

Provides new understanding of path structure in branching processes

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 $\frac{3}{2}\ln n$, provided the non-extinction 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 normalizatoin, converges weakly to a Brownian excursion in $D([0,1],\r)$.