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For a family of integer-valued height functions defined over the faces ofplanar graphs, we establish a relation between the probability of connection bylevel sets and the spin-spin correlations of the dual $O(2)$ symmetric spinmodels formulated over the graphs' vertices. The relation is used to show thatin two dimensions the Villain spin model exhibits non-summable decay ofcorrelations at any temperature at which the dual integer-restricted Gaussianfield exhibits depinning. For the latter, we devise a new monotonicity argumentthrough which the recent alternative proof by Lammers of the existence of adepinning transition in two-dimensional graphs of degree three, is extended toall doubly-periodic graphs, in particular to $\mathbb{Z}^2$. Essential use ismade of the inequality of Regev and Stephens-Davidowitz, which allows also analternative (to absolute-value FKG) proof of convergence of theheight-function's distribution in the infinite-volume limit. Similar resultsare established for the $XY$ spin model and its dual Bessel random heightfunction. Taken together these statements yield a new perspective on theBerezinskii-Kosterlitz-Thouless phase transition in $O(2)$ spin models, andcomplete a new proof of depinning in two-dimensional integer-valued heightfunctions.

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models