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

We consider dimer models on graphs which are bipartite, periodic and satisfy a geometric condition called isoradiality, defined in [18]. We show that the scaling limit of the height function of any such dimer model is 1/ √ π times a Gaussian free field. Triangular quadri-tilings were introduced in [6]; 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/ √ π times a Gaussian free field, and that the two Gaussian free fields are independent.