Quadri-tilings of the plane

We introduce quadri-tilings and show that they are in bijection with dimer models on a family of graphs R * arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called triangular quadri-tilings, as an interface model in dimension 2+2. Assigning “critical” weights to edges of R *, we prove an explicit expression, only depending on the local geometry of the graph R *, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of Kenyon (Invent Math 150:409–439, 2002). We also show that when edges of R * are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of all triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain. DOI: https://doi.org/10.1007/s00440-006-0002-9 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-21581 Accepted Version Originally published at: de Tilière, B (2007). Quadri-tilings of the plane. Probability Theory and Related Fields, 137(3-4):487518. DOI: https://doi.org/10.1007/s00440-006-0002-9 ar X iv :m at h/ 04 03 32 4v 1 [ m at h. PR ] 1 9 M ar 2 00 4 Quadri-tilings of the plane Béatrice de Tilière ∗ Abstract Quadri-tilings of the plane are tilings by quadrilaterals made of adjacent right triangles. Quadri-tilings are in bijection with dimer configurations on graphs arising from rhombus tilings of the plane. Assigning “critical” weights to the edges of such a graph, we construct a natural explicit Gibbs measure, and prove that it is asymptotically independent of the structure of the graph. We give an explicit expression for a measure on the set of dimer configurations of all graphs arising from 60◦-rhombus tilings of the plane, whose marginals are the above Gibbs measures. We construct two “height functions” on 60◦-rhombus quadri-tilings, and thereby interpret them as surfaces in a 4-dimensional space.Quadri-tilings of the plane are tilings by quadrilaterals made of adjacent right triangles. Quadri-tilings are in bijection with dimer configurations on graphs arising from rhombus tilings of the plane. Assigning “critical” weights to the edges of such a graph, we construct a natural explicit Gibbs measure, and prove that it is asymptotically independent of the structure of the graph. We give an explicit expression for a measure on the set of dimer configurations of all graphs arising from 60◦-rhombus tilings of the plane, whose marginals are the above Gibbs measures. We construct two “height functions” on 60◦-rhombus quadri-tilings, and thereby interpret them as surfaces in a 4-dimensional space.