Massless Phases for the Villain model in $d\geq 3$

We consider the classical Villain rotator model in $\mathbb{Z}^d, d\geq 3$ at sufficiently low temperature, and prove that the truncated two-point function decays asymptotically as $|x|^{2-d}$, with an algebraic rate of convergence. We also obtain the same asymptotic decay separately for the transversal two-point functions. This quantifies the spontaneous magnetization result for the Villain model at low temperature, and rigorously establishes the Gaussian spin-wave conjecture in dimension $d\ge 3$. We believe that our method extends to finite range interactions and to other abelian spin systems and abelian gauge theory in $d\geq 3$.