Double tangent method for two-periodic Aztec diamonds

TL;DR

This paper introduces a novel approach using the octahedron recurrence and tangent methods to compute refined partition functions and derive the arctic curve in two-periodic Aztec diamonds, advancing understanding of phase separation.

Contribution

It develops a new geometric tangent method and applies the octahedron recurrence to explicitly compute boundary refined partition functions and the arctic curve.

Findings

Derived the parametric form of the arctic curve separating phases.

Computed boundary two-refined partition functions exactly.

Reproduced the known algebraic equation of the arctic curve of degree 8.

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 recently 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.