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In this paper we investigate the height field of a dimer model/random dominotiling on the plane at a smooth-rough (i.e. gas-liquid) transition. We provethat the height field at this transition has two-point correlation functionswhich limit to those of the massless sine-Gordon field at the free fermionpoint, with parameters $(4\pi, z)$ where $z\in \mathbb{R}\setminus \{0\}$. Thedimer model is on $\epsilon \mathbb{Z}^2$ and has a two-periodic weightstructure 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, weprovide a direct asymptotic analysis of a double contour integral formula ofthe correlation kernel of the dimer model found by Fourier analysis. Thelimiting field interpolates between the Gaussian free field and white noise andthe main result gives an explicit connection between tiling/dimer models andthe law of a two-dimensional non-Gaussian field.

Two-periodic weighted dominos and the sine-Gordon field at the free fermion point: I