## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 6, 20 pp.

### One-point localization for branching random walk in Pareto environment

Marcel Ortgiese and Matthew I. Roberts

#### 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.

#### Article information

**Source**

Electron. J. Probab. Volume 22 (2017), paper no. 6, 20 pp.

**Dates**

Received: 29 April 2016

Accepted: 22 December 2016

First available in Project Euclid: 17 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1484622023

**Digital Object Identifier**

doi:10.1214/16-EJP22

**Subjects**

Primary: 60K37: Processes in random environments

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

branching random walk random environment parabolic Anderson intermittency localization

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Ortgiese, Marcel; Roberts, Matthew I. One-point localization for branching random walk in Pareto environment. Electron. J. Probab. 22 (2017), paper no. 6, 20 pp. doi:10.1214/16-EJP22. https://projecteuclid.org/euclid.ejp/1484622023