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Universal criterion for cutoff in non-reversible Markov processes

Develop a universal, model-independent criterion that characterizes when families of non-reversible or, more generally, non-normal Markov processes exhibit the cutoff phenomenon, analogous to the product-condition characterizations available in reversible or normal settings.

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Background

The paper presents a unifying framework for cutoff under a broad class of divergences in reversible and normal settings, establishing equivalences and product-type conditions (e.g., involving spectral gap and mixing times).

In contrast, for non-reversible or non-normal generators the authors provide partial progress via perturbation-based results and comparisons, but they note that existing results are largely model-specific and do not yield a general characterization. Hence, a universal criterion remains an open objective.

References

However, current results about cutoff phenomenon under non-reversible setting are still at primary stage, most of which only deal with some specific models, and a universal criterion is still quite open.

— Information-theoretic classification of the cutoff phenomenon in Markov processes (2407.06982 - Wang et al., 9 Jul 2024) in Section 4, Non-reversible cases (first paragraph)