Double tangent method for two-periodic Aztec diamonds

Abstract

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.

Arctic curves of the T-system with slanted initial data

“…This formulation has the advantage of giving an alternative route to access to thermodynamic properties of the models, and in particular the arctic phenomenon: we may hope to be able to use the so-called tangent method of Colomo and Sportiello [9-11, 13-17, 21], and compare the results to those obtained in the present paper. Some advances in this direction were perfomed in [39] for the case of the two-periodic Aztec diamond.…”