It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits depinning. For the latter, we extend the recent proof by Lammers of the existence of a depinning transition in the integer-valued Gaussian field in two-dimensional cubic graphs to all doubly periodic graphs, in particular to Z^2. Taken together these two statements yield a new perspective on the Berezinskii–Kosterlitz–Thouless phase transition in the Villain model, and complete a new proof of depinning in two-dimensional integer-valued height functions.

Based on joint work with: Michael Aizenman, Matan Harel and Ron Peled.