Scaling limit and ageing for branching random walk in Pareto environment

TL;DR

This paper studies a branching random walk in a Pareto environment, demonstrating a scaling limit involving 'lilypads' and showing the system's maximizer exhibits ageing, revealing new insights into such stochastic processes.

Contribution

It introduces a novel scaling limit for branching random walk in Pareto environments and characterizes the limiting structure as 'lilypads' built on a Poisson process.

Findings

Convergence to a 'lilypads' structure in the scaling limit

Demonstration of ageing property in the maximizer

Description of the limit object as a Poisson-based growth process

Abstract

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of "lilypads" built on a Poisson point process in $\mathbb{R}^d$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.