Conformal invariance of isoradial dimer models & the case of triangular quadri-tilings

Abstract

We consider dimer models on graphs which are bipartite, periodic and satisfy a geometric condition called {\em isoradiality}, defined in \cite{Kenyon3}. We show that the scaling limit of the height function of any such dimer model is $1/\sqrt{\pi }$ times a Gaussian free field. Triangular quadri-tilings were introduced in \cite{Bea}; they are dimer models on a family of isoradial graphs arising form rhombus tilings. By means of two height functions, they can be interpreted as random interfaces in dimension 2+2. We show that the scaling limit of each of the two height functions is $1/\sqrt{\pi }$ times a Gaussian free field, and that the two Gaussian free fields are independent.

32 pages, 4 figures

References in corpus (3)

• Local statistics of lattice dimers
• An introduction to the dimer model
• Discrete period matrices and related topics

Cited by in corpus (8)

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• Asymptotics of height change on toroidal Temperleyan dimer models
• Central limit theorem for lozenge tilings with curved limit shape
• Conformal invariance of dimer heights on isoradial double graphs
• Derivation of field theory for the classical dimer model using bosonization