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.