Tangent method for the arctic curve arising from freezing boundaries

In the paper Di Francesco and Guitter (2018 J. Phys. A: Math. Theor. 51 355201), the authors study the arctic curve arising in random tilings of some planar domains with an arbitrary distribution of defects on one edge. Using the tangent method they derive a parametric equation for portions of arctic curve in terms of an arbitrary piecewise differentiable function that describes the defect distribution. When this distribution presents 'freezing' intervals, other portions of arctic curve appear and typically have a cusp. These freezing boundaries can be of two types, respectively with maximal or minimal density of defects. Our purpose here is to extend the tangent method derivation of Di Francesco and Guitter (2018 J. Phys. A: Math. Theor. 51 355201) to include these portions, hence answering the open question stated in Di Francesco and Guitter (2018 J. Phys. A: Math. Theor. 51 355201). (paper: quantum statistical physics, condensed matter, integrable systems)