Arctic Curves in path models from the Tangent Method

TL;DR

This paper applies the Tangent Method to compute arctic curves in various lattice path models, recovering known results and discovering new shapes, including arctic half-ellipse and parabola, in different tiling and path configurations.

Contribution

It extends the Tangent Method to models with osculating contact points and provides asymptotic analysis of one-point functions in these models.

Findings

Recovered the arctic circle in domino tilings of the Aztec diamond.

Discovered an arctic half-ellipse in a Dyck path model.

Identified an arctic parabola in a rhombus tiling model.

Abstract

Recently, Colomo and Sportiello introduced a powerful method, known as the \emph{Tangent Method}, for computing the arctic curve in statistical models which have a (non- or weakly-) intersecting lattice path formulation. We apply the Tangent Method to compute arctic curves in various models: the domino tiling of the Aztec diamond for which we recover the celebrated arctic circle; a model of Dyck paths equivalent to the rhombus tiling of a half-hexagon for which we find an arctic half-ellipse; another rhombus tiling model with an arctic parabola; the vertically symmetric alternating sign matrices, where we find the same arctic curve as for unconstrained alternating sign matrices. The latter case involves lattice paths that are non-intersecting but that are allowed to have osculating contact points, for which the Tangent Method was argued to still apply. For each problem we estimate the large size asymptotics of a certain one-point function using LU decomposition of the corresponding Gessel-Viennot matrices, and a reformulation of the result amenable to asymptotic analysis.