Develop a mode-ordered renormalisation group strategy

Develop a renormalisation group framework that explores interaction modes in an ordered fashion (analogous to revealing scales from smallest to largest in the Polchinski flow) to potentially relax the need for simultaneous strong convexity across all modes in the criterion for uniform log-Sobolev inequalities.

Background

The paper’s criterion for uniform log-Sobolev inequalities imposes strong convexity of the projected free energy \hat{\mathcal F}_T across all modes simultaneously, akin to Bakry–Émery conditions. This contrasts with the renormalisation group approach used in short-range models via Polchinski’s flow, which progressively integrates scales from small to large.

The authors suggest exploring a mode-by-mode renormalisation group strategy, potentially offering a more refined analysis tailored to the modal structure of mean-field interactions.