Double tangent method for two-periodic Aztec diamonds

• Published in: Journal of Statistical Mechanics: Theory and Experiment
• Main Author: Ruelle, Philippe
• Other Authors: UCL - SST/IRMP - Institut de recherche en mathématique et physique
• Format: Article in Journal/Newspaper
• Language: English
• Published: Institute of Physics Publishing Ltd. 2022
• Subjects: arctic curves; tangent method; domino tilings; Aztec diamonds; octahedron recurremce; Arctic
• Online Access: http://hdl.handle.net/2078.1/268643; https://doi.org/10.1088/1742-5468/aca4c4

We use the octahedron recurrence, which generalizes the quadratic recurrence found by Kuo for standard Aztec diamonds, in order to compute boundary one-refined and two-refined partition functions for two-periodic Aztec diamonds. In a first approach, the geometric tangent method allows to derive the parametric form of the arctic curve, separating the solid and liquid phases. This is done by using the recent reformulation of the tangent method and the one-refined partition functions without extension of the domain. In a second part, we use the two-refined tangent method to rederive the arctic curve from the boundary two-refined partition functions, which we compute exactly on the lattice. The curve satisfies the known algebraic equation of degree 8, of which either tangent method gives an explicit parametrization.