On some generalized reinforced random walks on integers

TL;DR

This paper investigates generalized reinforced random walks on integers, establishing conditions for recurrence or transience and identifying a phase transition phenomenon similar to that observed on trees.

Contribution

It introduces new conditions for recurrence and transience in reinforced random walks and demonstrates a phase transition analogous to prior work on trees.

Findings

Conditions for recurrence and transience are derived.

A phase transition phenomenon is identified.

Similarities to phase transitions on trees are shown.

Abstract

We consider Reinforced Random Walks where transition probabilities are a function of the proportion of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle \cite{Pem000} on trees.