On the maximal displacement of catalytic branching random walk

TL;DR

This paper analyzes the maximal displacement distribution in catalytic branching random walks on Z, revealing heavy tail behaviors that differ from classical branching random walks, especially in critical and subcritical regimes.

Contribution

It introduces new effects in the maximal displacement distribution for catalytic branching random walks, highlighting differences from standard models.

Findings

Maximal displacement has a heavy tail decreasing as a power of 1/2 in critical cases.

In subcritical cases, the tail decreases as a power of 1.

New phenomena are identified that are absent in classical branching random walks.

Abstract

We study the distribution of the maximal displacement of particles positions for the whole time of the population existence in the model of critical and subcritical catalytic branching random walk on Z. In particular, we prove that in the case of simple symmetric random walk on Z, the distribution of the maximal displacement has "a heavy tail" decreasing as a function of the power 1/2 or 1, when the branching process is critical or subcritical, respectively. These statements describe new effects which do not arise in the corresponding investigations of the maximal displacement of critical and subcritical branching random walks on Z.