Cutoff Phenomenon for Cyclic Dynamics on Hypercube

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.