Cookie branching random walks

TL;DR

This paper studies a branching random walk model on the integer line where particles' behavior depends on whether sites have been visited before, analyzing conditions for recurrence or transience of the process.

Contribution

It introduces a cookie-dependent branching random walk model and investigates its recurrence and transience properties based on site visitation history.

Findings

The process's recurrence or transience depends on initial cookie configuration.

Conditions under which particles visit the origin infinitely often are characterized.

Behavior differs significantly from classical branching random walks without cookies.

Abstract

We consider a branching random walk on $\Z$, where the particles behave differently in visited and unvisited sites. Informally, each site on the positive half-line contains initially a cookie. On the first visit of a site its cookie is removed and particles at positions with a cookie reproduce and move differently from particles on sites without cookies. Therefore, the movement and the reproduction of the particles depend on the previous behaviour of the population of particles. We study the question if the process is recurrent or transient, i.e., whether infinitely many particles visit the origin or not.