  For a family of integer-valued height functions defined over the faces of
planar graphs, we establish a relation between the probability of connection by
level sets and the spin-spin correlations of the dual $O(2)$ symmetric spin
models formulated over the graphs' vertices. The relation is used to show that
in two dimensions the Villain spin model exhibits non-summable decay of
correlations at any temperature at which the dual integer-restricted Gaussian
field exhibits depinning. For the latter, we devise a new monotonicity argument
through which the recent alternative proof by Lammers of the existence of a
depinning transition in two-dimensional graphs of degree three, is extended to
all doubly-periodic graphs, in particular to $\mathbb{Z}^2$. Essential use is
made of the inequality of Regev and Stephens-Davidowitz, which allows also an
alternative (to absolute-value FKG) proof of convergence of the
height-function's distribution in the infinite-volume limit. Similar results
are established for the $XY$ spin model and its dual Bessel random height
function. Taken together these statements yield a new perspective on the
Berezinskii-Kosterlitz-Thouless phase transition in $O(2)$ spin models, and
complete a new proof of depinning in two-dimensional integer-valued height
functions.
