Scaling limits and ergodic Gibbs measures at nonzero tilt

Determine whether an ergodic gradient Gibbs measure exists for general nonzero tilt and non-convex interaction potentials, and, where such measures exist, characterize the conditional mean of the field given the disorder variables in order to establish the corresponding scaling limit.

Background

The paper proves a Gaussian Free Field scaling limit only for shift-invariant ergodic tempered gradient Gibbs measures with zero tilt. Extending the result to nonzero tilt would require understanding the mean of the field conditioned on the disorder variables in the mixture representation. The authors note that this difficulty was already identified in the Gaussian setting and is more severe for the non-Gaussian disordered models considered here.

The existence of an ergodic gradient Gibbs measure with a prescribed nonzero tilt is itself unresolved for general non-convex potentials. Thus, both the existence question and the analysis needed for a nonzero-tilt scaling limit remain open.

References

The case of nonzero tilt seems also very hard. In that case one would need to understand the mean of the field conditioned on the κ. Already in the Gaussian case this was called a “hard open problem” in [15], and the non-Gaussian case here seems even harder. Additionally, for general non-zero tilt and non-convex V it is not even clear whether there exists an ergodic gradient Gibbs measure with that tilt, and answering this question is another hard open problem in the area.

— Gradient Gibbs measures with non-convex potentials and the universality class of the Gaussian Free Field  (2608.14526 - Buchholz et al., 14 Aug 2026) in Section 1.3, “Scaling limit in other settings”