Maximal support of real-space renormalization group transformations

Determine the maximal support of any real-space renormalization group transformation on a suitable space of well-behaved interactions, including whether that support contains the critical point of the Ising model or general systems at criticality.

Background

The paper distinguishes between the well-defined action of a real-space renormalization group transformation on probability measures and the more restrictive question of whether the transformed measures remain compatible with sufficiently regular interactions. Previous work cited by the paper shows that this compatibility can fail, including for the nearest-neighbour Ising model in part of the low-temperature regime. The unresolved problem is to characterize the largest domain on which the interaction-level transformation is well defined and to establish whether critical systems belong to that domain.

References

It remains an open question to determine the maximal support of $R_T$ within $\mathscr{V}$ for any given $T$ and whether such a map is well-defined for models at a second order phase transition.

— Pathology-Free Real-Space Renormalization Group Theory on an Inverse Limit Space  (2609.18356 - Arz, 16 Sep 2026) in Section 1, Introduction

It remains an open question, to find the maximal support of any real-space RGT on a suitable space of well-behaved interactions and whether this will include the critical point of the Ising model/general systems at criticality 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 1, subsection “Real-Space Renormalization Group Transformations”