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

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

2005-12-16

arxiv.org/abs/math/0512395v1

Annales de l'institut Henri Poincar\'e (B) Probabilit\'es et Statistiques, 43 no. 6 (2007), p. 729-750

Mathematics

Probability

32 pages, 4 figures

Scientific paper

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.

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