Determine whether ergodic gradient Gibbs measures with interior slope have finite specific free energy

Determine whether every ergodic gradient Gibbs measure with slope in the interior of the admissible slope set has finite specific free energy.

Background

Specific free energy is used to distinguish minimal gradient Gibbs measures and to formulate the variational theory of convex gradient models. The cited conjecture concerns whether ergodicity and an interior slope alone guarantee that the specific free energy is finite.

The paper notes that this property is conjectured in the general theory and does not establish it for arbitrary interactions. It is known in some special settings, including Lipschitz height functions and isotropic interactions, as part of the variational principle.

References

Conversely,Conjecture 10.3.2 conjectures that if $\mu\nabla$ is ergodic and $slope(\mu\nabla)\in \mathfrak{S}\circ$ then $SFE(\mu\nabla)<\infty$ (the restriction to the interior of $\mathfrak{S}$ may not be relevant in our integer-valued setup).

— On the rigidity of sloped height functions in $d\ge 3$ and non-crossing surfaces  (2609.01499 - Adhikari et al., 1 Sep 2026) in Section 2, Subsection 2.4, Infinite free energy and non-minimality