Cutoff Phenomenon for Cyclic Dynamics on Hypercube

TL;DR

This paper investigates the cutoff phenomenon in certain irreversible cyclic Markov chains on hypercubes, providing a comprehensive proof and extending understanding beyond reversible models.

Contribution

It introduces a detailed analysis of cutoff phenomena for irreversible cyclic dynamics on hypercubes, expanding the scope of existing Markov chain cutoff studies.

Findings

Established cutoff behavior for specific irreversible cyclic dynamics.

Extended cutoff theory to non-reversible Markov chains.

Provided rigorous proof based on coupling modifications.

Abstract

The cutoff phenomena for Markovian dynamics have been observed and rigorously verified for a multitude of models, particularly for Glauber-type dynamics on spin systems. However, prior studies have barely considered irreversible chains. In this work, the cutoff phenomenon of certain cyclic dynamics are studied on the hypercube $\Sigma_{n} = Q^{V_{n}}$, where $Q = \{1, 2, 3\}$ and $V_{n} = \{1,...,n\}$. The main feature of these dynamics is the fact that they are represented by an irreversible Markov chain. Based on the coupling modifications suggested in a previous study of the cutoff phenomenon for the Curie-Weiss-Potts model, a comprehensive proof is presented.