Locally Markov walks on finite graphs

TL;DR

This paper introduces locally Markov walks as a generalization of Markov chains, analyzes their properties, and studies the mixing time of a specific case, demonstrating cutoff behavior.

Contribution

It defines locally Markov walks, explores their fundamental properties, and analyzes the mixing time of the uniform unicycle walk on complete graphs, showing cutoff.

Findings

Stationary distribution and recurrent states characterized

Irreducibility and ergodicity established

Uniform unicycle walk exhibits cutoff in mixing time

Abstract

Locally Markov walks are natural generalizations of classical Markov chains, where instead of a particle moving independently of the past, it decides where to move next depending on the last action performed at the current location. We introduce the concept of locally Markov walks and we describe their stationary distribution and recurrent states, and we prove several properties such as irreducibility and ergodicity. For a particular locally Markov walk - the uniform unicycle walk on the complete graph - we investigate the mixing time and we prove that it exhibits cutoff.