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Two-periodic weighted dominos and the sine-Gordon field at the free fermion point: I
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abstract
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.
Forward citations
Cited by 3 Pith papers
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Twisted Dirac operators and fractional correlations of the massless sine-Gordon model at the free fermion point
Fractional vertex-operator correlations of the massless sine-Gordon model at β = 4π are shown to equal Palmer's tau functions of massive twisted Dirac operators, giving a proof of the Lukyanov-Zamolodchikov one-point formula.
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Quenched and Annealed CLTs for the one-periodic Aztec diamond in random environment
Quenched height fluctuations in random-environment one-periodic Aztec diamonds converge almost surely to the Gaussian Free Field; annealed fluctuations are Gaussian with environment-dependent covariances.
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Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions
For Aztec diamond tilings with i.i.d. one-periodic edge weights, height function fluctuations are, in the critical regime, GFF plus independent Brownian motion, and in the fixed-variance regime, Brownian motion alone ...
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