The Spread of a Catalytic Branching Random Walk

TL;DR

This paper studies a branching random walk on the integers where branching occurs only at the origin, establishing a law of large numbers for the maximum position and classifying possible limiting distributions.

Contribution

It introduces a detailed analysis of the maximal position in a catalytic branching random walk, including laws of large numbers and limit laws for the maximum.

Findings

Law of large numbers for the maximal position $M_n$ in the supercritical regime

Complete classification of limiting laws for $M_n - eta n$

Identification of the deterministic speed $eta$ in the model

Abstract

We consider a random walk on $\Z$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$. Then we determine all possible limiting law for the sequence $M_n -\alpha n$ where $\alpha$ is a deterministic constant.