Random walks on primitive lattice points

TL;DR

This paper studies a random walk on primitive lattice points in integer space, proving that under mild conditions, these walks are strongly recurrent with a unique stationary distribution.

Contribution

It introduces a new random walk model on primitive lattice points and establishes its positive recurrence and uniqueness of stationary measure.

Findings

The random walk is positive recurrent.

There exists a unique stationary measure.

The walk exhibits strong recurrence properties.

Abstract

We define a random walk on the set of primitive points of $\mathbb{Z}^d$. We prove that for walks generated by measures satisfying mild conditions these walks are recurrent in a strong sense. That is, we show that the associated Markov chains are positive recurrent and there exists a unique stationary measure for the random walk.