Extension of Ising RG-flow arguments to a three-state Potts cluster construction

Determine whether the connectivity and variance arguments developed for the nearest-neighbour Ising model also apply to the three-state Potts-model renormalization group construction that incorporates horizontal bonds within renormalization-group blocks.

Background

The paper proves that, for Potts models with at least three states, non-linear transformations cannot generally be represented using only vertical cluster bonds. It then presents a construction for a three-state Potts model using additional horizontal bonds within blocks, which permits a broader family of transformations. Whether the analytical arguments used for the Ising model remain valid for this modified cluster representation is left unresolved.

References

Nevertheless, it remains an open question whether the arguments we have used for the Ising model also apply in such a construction or not.

— Cluster Representation of Renormalization Group Transformations and a Rigorous Proof for Convergence of the RG-Flow of the Ising Model to Trivial Fixed Points away from Criticality  (2608.18862 - Arz, 19 Aug 2026) in Section 5, subsection “Generalizations”