Long range one-cookie random walk with positive speed

TL;DR

This paper investigates a one-dimensional excited random walk with long-range jumps, establishing conditions under which the walk exhibits positive speed due to the interplay of excited and symmetric behaviors.

Contribution

It introduces a model with non-nearest neighbor jumps and provides a sufficient condition for the walk to have positive speed, extending previous excited random walk results.

Findings

The walk can have positive speed under certain conditions.

Long-range jumps influence the speed and behavior of the walk.

The model combines excited and symmetric random walk features.

Abstract

We study one-dimensional excited random walks with non-nearest neighbor jumps. When the process is at a vertex that has not been visited before, its next transition has a positive drift to the right, possibly with long jumps. Whenever the process visits a vertex that has already been visited in the past, its next transition is the one of a simple symmetric random walk. We give a sufficient condition for the process to have positive speed.