Locally Markov walks on finite graphs

TL;DR

This work introduces locally Markov walks as a natural generalization of Markov chains on finite graphs, where the next move depends on the last action at the current vertex. It builds the total chain to encode local memory, proves existence and ergodicity of stationary distributions governed by $q_x$—the local chain’s stationary distributions—and reveals that recurrent states correspond to spanning unicycles with a $q$-weighted measure. The authors then specialize to the uniform unicycle walk on complete graphs, deriving its spectral structure and proving a sharp cutoff at $m\log m$ via connections to spanning trees and coupon collector arguments. The framework unifies rotor, excited, and tree-walk models, and provides a tractable approach to analyze mixing times and recurrence in generalized memory-dependent walks.

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.

Figures (2)

• Figure 1: Illustration of a step of the uniform unicycle walk on the complete graph with $3$ vertices.
• Figure 2: The unicycle graph $\mathsf{UCYC}(K_3)$ together with the uniform unicycle walk. The vertex colored in red indicates the particle location.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Example 2.1
• Example 2.2
• Example 2.3
• Remark 2.1
• Example 2.4
• Definition 2.2: Total chain associated to a locally Markov walk
• Remark 2.2