  In this paper we investigate the height field of a dimer model/random domino
tiling on the plane at a smooth-rough (i.e. gas-liquid) transition. We prove
that the height field at this transition has two-point correlation functions
which limit to those of the massless sine-Gordon field at the free fermion
point, with parameters $(4\pi, z)$ where $z\in \mathbb{R}\setminus \{0\}$. The
dimer model is on $\epsilon \mathbb{Z}^2$ and has a two-periodic weight
structure with weights equal to either 1 or $a=1-C|z|\epsilon$, for
$0<\epsilon$ small (tending to zero). In order to obtain this result, we
provide a direct asymptotic analysis of a double contour integral formula of
the correlation kernel of the dimer model found by Fourier analysis. The
limiting field interpolates between the Gaussian free field and white noise and
the main result gives an explicit connection between tiling/dimer models and
the law of a two-dimensional non-Gaussian field.
