Cutoff Phenomenon for Cyclic Dynamics on Hypercube

Keunwoo Lim

2022, v.28, Issue 1, 87-112


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 $V_{n} = \{1,\ldots ,n\}$. The main
feature of these dynamics is the fact that they are represented by an
irreversible Markov chain. Based on the couplings modified from the previous study of the cutoff phenomenon for the Curie-Weiss-Potts
model, a comprehensive proof is presented.

Keywords: Irreversible Markov chain, mixing time, cutoff phenomenon


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