A balanced excited random walk

TL;DR

This paper investigates a novel random walk on , where the walk's behavior switches between the first two and last two coordinates upon visits, and proves that this process is almost surely transient.

Contribution

It introduces a new type of excited random walk on  with coordinate-dependent behavior and proves its transience, expanding understanding of such processes.

Findings

The process is almost surely transient in .

Lower-dimensional cases are discussed.

Various generalizations are proposed.

Abstract

The following random process on $\Z^4$ is studied. At first visit to a site, the two first coordinates perform a (2-dimensional) simple random walk step. At further visits, it is the last two coordinates which perform a simple random walk step. We prove that this process is almost surely transient. The lower dimensional versions are discussed and various generalizations and related questions are proposed.