One-point localization for branching random walk in Pareto environment

TL;DR

This paper studies a branching random walk in a Pareto environment, demonstrating that most particles concentrate at a single high-potential site, with detailed estimates supporting this one-point localization phenomenon.

Contribution

It establishes a strong form of intermittency and one-point localization for branching random walk in Pareto environments, extending known results from the parabolic Anderson model.

Findings

Most mass concentrates at a single site with high potential

High probability of one-point localization

Supports previous small island concentration results

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 a very strong form of intermittency, where with high probability most of the mass of the system is concentrated in a single site with high potential. The analogous one-point localization is already known for the parabolic Anderson model, which describes the expected number of particles in the same system. In our case, we rely on very fine estimates for the behaviour of particles near a good point. This complements our earlier results that in the rescaled picture most of the mass is concentrated on a small island.