Depinning in the integer-valued Gaussian Field and the BKT phase of the 2D Villain model

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 P. Lammers of the existence of a depinning transition in the integer valued Gaussian field in two dimensional graphs of degree three to all doubly periodic graphs, in particular to $\mathbb{Z}^2$. Taken together these two statements yield a new perspective on the BKT phase transition in the Villain model, and complete a new proof of depinning in two dimensional integer-valued height functions.