Intermittency for branching random walk in Pareto environment

TL;DR

This paper studies a branching random walk in a Pareto-distributed potential, revealing its intermittent nature and contrasting its growth mechanisms with the parabolic Anderson model.

Contribution

It introduces a detailed shape theorem for the process and demonstrates the intermittency phenomenon in a Pareto environment.

Findings

Most particles concentrate on a small island with high potential.

The process exhibits a shape theorem described by growing lilypads.

Growth mechanisms differ from those in the parabolic Anderson model.

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 describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.