Twoperiodic weighted dominos and the sineGordon field at the free fermion point: I
Abstract
In this paper we investigate the height field of a dimer model/random domino tiling on the plane at a smoothrough (i.e. gasliquid) transition. We prove that the height field at this transition has twopoint correlation functions which limit to those of the massless sineGordon 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 twoperiodic weight structure with weights equal to either 1 or $a=1Cz\epsilon$, 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 twodimensional nonGaussian field.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 DOI:
 10.48550/arXiv.2209.11111
 arXiv:
 arXiv:2209.11111
 Bibcode:
 2022arXiv220911111M
 Keywords:

 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 Fixed misprints and the statement of Theorem 4 following some comments