Tree Builder Random Walk: recurrence, transience and ballisticity

TL;DR

This paper studies a novel class of random walks on growing trees, revealing conditions under which the walk is recurrent or transient, and identifying when it exhibits ballistic behavior based on the growth parameter s.

Contribution

It introduces the Tree Builder Random Walk model and characterizes its recurrence, transience, and ballisticity depending on the growth parameter s.

Findings

For odd s, the walk is ballistic and transient.

For even s, the walk can be null recurrent or trapped.

The walk's behavior depends on local properties of the growing tree.

Abstract

The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of markovian and non-markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or transient. If s is odd, the walker is ballistic and thus transient. If s is even, the walker's behavior can be explained from local properties of the growing tree and it can be either null recurrent or it gets trapped on some limited part of the growing tree.