Deterministically driven random walks on a finite state space

TL;DR

This paper introduces deterministic walks on finite state spaces, establishing conditions for transitivity, robustness, and the existence of asymptotic occupation times, thereby advancing understanding of deterministic dynamical systems.

Contribution

It provides new hypotheses and conditions that guarantee transitivity and asymptotic occupation times in deterministic walks on finite spaces, highlighting their robustness.

Findings

Deterministic walks can be transitive under certain hypotheses.

Transitivity of deterministic walks is robust to perturbations.

Conditions for the existence of asymptotic occupation times are established.

Abstract

We introduce the concept of a deterministic walk. Confining our attention to the finite state case, we establish hypotheses that ensure that the deterministic walk is transitive, and show that this property is in some sense robust. We also establish conditions that ensure the existence of asymptotic occupation times.