It is shown that the Villain model of two-component spinsover two dimensional lattices exhibits slow, non-summable, decay ofcorrelations at any temperature at which the dual integer-valued Gaussian fieldexhibits delocalization. For the latter, we extend the recent proof by Lammersof a delocalization transition in two dimensional graphs of degree three, toall doubly periodic graphs, in particular to Z^2. Taken together these twostatements yield a new perspective on the BKT phase transition in the Villainmodel, and a new proof on delocalization in two dimensional integer-valuedheight functions. Joint with Aizenman, Harel and Peled.

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