Removing the uniform upper bound on the component curvatures

Establish the scaling-limit theory for strongly log-concave log-mixtures whose strictly convex component potentials have no uniform upper bound on their second derivatives, including the required strong solutions, time ergodicity, and extremality of the associated disordered Gibbs measures.

Background

The decomposition theorem can produce strongly log-concave log-mixtures with component growth controlled by a function related to the original potential’s second derivative, even when the original second derivative is not uniformly bounded above. Several estimates and homogenization inputs appear compatible with this broader setting.

The main obstruction is dynamical: without a uniform upper bound on the component curvatures, the Langevin drifts are not globally Lipschitz, so the strong-solution argument used for uniqueness and time ergodicity no longer applies directly. The authors also identify extremality of fixed-disorder Gibbs measures as a difficult unresolved issue.

References

These SDEs still have weak solutions (as follows from [5, Theorem 4.4]), but proving existence of strong solutions seems very difficult. In the absence of that, it is unclear how to obtain time-ergodicity of the corresponding dynamics. Via [5, Theorem 5.15] it would follow if we knew that for a.e. κ the disordered Gibbs measure is extremal, but extremality of Gibbs measures is a notoriously hard question as well (and as we need extremality for fixed κ we are not in any translation-invariant setting, so the abstract theory of [63] does not apply). While we currently do not know how to resolve this problem, the discussion above served as our motivation to point out that if we have a strongly log-concave log-mixture of quadratic growth then the corresponding disordered Gibbs measures are in fact a.s. extremal (see Theorem 6.4).

— 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, “Extensions to other interface models—More degenerate potentials”