Convergence in law for the branching random walk seen from its tip

TL;DR

This paper extends recent methods to prove that the entire point process of a critical branching random walk, viewed from its tip, converges in law to a Poisson point process, confirming a conjecture and paralleling results for branching Brownian motion.

Contribution

It adapts Aidekon's method to show convergence of the full point process of the branching random walk from its tip, confirming a conjecture and providing a discrete analog to continuous models.

Findings

Point process of the branching random walk converges to a Poisson point process.

Results confirm a conjecture of Brunet and Derrida.

Analogous to known results for branching Brownian motion.

Abstract

Considering a critical branching random walk on the real line. In a recent paper, Aidekon [3] developed a powerful method to obtain the convergence in law of its minimum after a log-factor normalization. By an adaptation of this method, we show that the point process formed by the branching random walk and its minimum converge in law to a Poisson point process colored by a certain point process. This result, confirming a conjecture of Brunet and Derrida [10], can be viewed as a discrete analog of the corresponding results for the branching brownian motion, previously established by Arguin et al. [5] [6] and Aidekon et al. [2].