Scaling limit of isoradial dimer models and the case of triangular quadri-tilings

Abstract

We consider dimer models on planar graphs which are bipartite, periodic and satisfy a geometric condition called isoradiality, defined in [R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math. 150 (2) (2002) 409–439]. We show that the scaling limit of the height function of any such dimer model is a Gaussian free field. Triangular quadri-tilings were introduced in [B. de Tilière, Quadri-tilings of the plane, math.PR/0403324, Probab. Theory Related Fields, in press]; they are dimer models on a family of isoradial graphs arising from 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 a Gaussian free field, and that the two Gaussian free fields are independent.