Point process convergence for branching random walks with regularly varying steps

TL;DR

This paper studies the limiting behavior of point processes in a branching random walk with regularly varying steps, showing convergence to a Cox cluster process and confirming a conjecture by Brunet and Derrida.

Contribution

It establishes the weak convergence of scaled displacements to a Cox cluster process and confirms a conjecture in the context of branching random walks with regularly varying steps.

Findings

Convergence of point processes to Cox cluster process

Validation of Brunet and Derrida's conjecture

Recovery of Durrett's main result in this framework

Abstract

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of Durrett (1983) in our framework.