A note on multi-type cookie random walk on integers

TL;DR

This paper analyzes a multi-type cookie random walk on integers, showing that despite initial positive drift, the walk is almost surely transient to the left with positive speed due to changing drift after visits.

Contribution

It introduces a model with changing drift based on visit counts and proves the walk's transience and positive speed to the left.

Findings

Walker is almost surely transient to the left

The walk has positive speed despite initial positive drift

Changing drift influences long-term behavior

Abstract

We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to the left with positive speed.