Tangent method for the arctic curve arising from freezing boundaries

TL;DR

This paper extends the tangent method to derive the arctic curve in random tilings with freezing boundaries, revealing new cusp-shaped portions and addressing an open question from prior work.

Contribution

It introduces an extension of the tangent method to include freezing boundary effects in arctic curves, capturing cusp formations and complex boundary behaviors.

Findings

Derived parametric equations for arctic curves with freezing boundaries

Identified cusp formations at freezing boundaries

Extended the tangent method to new boundary conditions

Abstract

In the paper arXiv:1803.11463, 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 arXiv:1803.11463 to include these portions, hence answering the open question stated in arXiv:1803.11463.