Height function delocalisation on cubic planar graphs

The interest is in models of integer-valued height functions on shift-invariant planar graphs whose maximum degree is three. We prove delocalisation for models induced by convex nearest-neighbour potentials, under the condition that each potential function is an excited potential, that is, a convex symmetric potential function $V$ with the property that $V(\pm1)\leq V(0)+\log2$. Examples of such models include the discrete Gaussian and solid-on-solid models at inverse temperature $\beta\leq\log2$, as well as the uniformly random $K$-Lipschitz function for fixed $K\in\mathbb N$. In fact, $\beta V$ is an excited potential for any convex symmetric potential function $V$ whenever $\beta$ is sufficiently small. To arrive at the result, we develop a new technique for symmetry breaking, and then study the geometric percolation properties of sets of the form $\{\varphi\geq a\}$ and $\{\varphi\leq a\}$, where $\varphi$ is the random height function and $a$ a constant. Along the same lines, we derive delocalisation for models induced by convex symmetric nearest-neighbour potentials which force the parity of the height of neighbouring vertices to be distinct. This includes models of uniformly random graph homomorphisms on the honeycomb lattice and the truncated square tiling, as well as on the same graphs with each edge replaced by $N$ edges linked in series. The latter resembles cable-graph constructions which appear in the analysis of the Gaussian free field.