  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 its second
derivative satisfies $V''(0)\leq2\log2$. 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 directly 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 discrete Gaussian free field.
