  Height functions are random functions on a given graph, in our case
integer-valued functions on the two-dimensional square lattice. We consider
gradient potentials which (informally) lie between the discrete Gaussian and
solid-on-solid model (inclusive). The phase transition in this model, known as
the roughening transition, Berezinskii-Kosterlitz-Thouless transition, or
localisation-delocalisation transition, was established rigorously in the 1981
breakthrough work of Fr\"ohlich and Spencer. It was not until 2005 that
Sheffield derived continuity of the phase transition. First, we establish
sharpness, in the sense that covariances decay exponentially in the localised
phase. Second, we show that the model is delocalised at criticality, in the
sense that the set of potentials inducing localisation is open in a natural
topology. Third, we prove that the pointwise variance of the height function is
at least $c\log n$ in the delocalised regime, where $n$ is the distance to the
boundary, and where $c>0$ denotes a universal constant. This implies that the
effective temperature of any potential cannot lie in the interval $(0,c)$
(whenever it is well-defined), and jumps from $0$ to at least $c$ at the
critical point. We call this range of forbidden values the effective
temperature gap.
