4 problems

Let $U_{\phi}$ be the set appearing in the source, and let $\mu$ be a non-trivial gradient Gibbs measure, meaning that $\mu$-almost surely $Z_{\Lambda}(\phi)<\infty$ for every fini…

Let $U_{\Phi}$ be the set of interior slopes. A non-$\mathcal L$-invariant gradient Gibbs measure $\mu\in\mathcal P(\Omega,\mathcal F^{\tau})$ has approximate slope $u$ if, for eve…

Let $U_{\Phi}=\{u:|u_1|+|u_2|<\frac{1}{2}\}$ be the set of interior slopes, and let $\mu_u$ be the explicitly described ergodic gradient Gibbs measure of slope $u$. Uniqueness conj…

Smooth-phase conjecture. Every ergodic gradient Gibbs measure is in a smooth phase in every dimension $d\geq 3$; equivalently, each such measure is the restriction of a non-gradien…