5 years ago

Arctic Curves in path models from the Tangent Method.

Matthew F. Lapa, Philippe Di Francesco

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.

Publisher URL: http://arxiv.org/abs/1711.03182

DOI: arXiv:1711.03182v1

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