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Quadri-tilings of the Plane

de Tilière, Béatrice

In: Probability Theory and Related Fields, 2007, vol. 137, no. 3-4, p. 487-518

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    Title
    • Quadri-tilings of the Plane
    Author
    • de Tilière, Béatrice. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Probability Theory and Related Fields, 2007, vol. 137, no. 3-4, p. 487-518. Springer-Verlag
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:318998
    Summary
    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