Unified MLSI criterion

Develop a quantitative modified log-Sobolev inequality criterion that encompasses existing localization-based entropy criteria and the scalar-field hypotheses for ordinary matching Glauber dynamics, while avoiding losses caused by small occupied marginals.

Background

The paper establishes a log-Sobolev criterion based on field-dynamics spectral stability, marginal-ratio bounds, and occupied-marginal lower bounds. For ordinary matching Glauber dynamics, this yields an inverse MLSI bound with a logarithmic dependence on the inverse occupied-marginal lower bound.

The authors ask whether a broader quantitative criterion can unify several existing entropy methods and the scalar-field framework, and identify an O_λ(m) inverse MLSI bound as a benchmark that would improve the mixing-time dependence on n.