# Arctic Curves In Path Models from The Tangent Method

Abstract : Recently, Colomo and Sportiello introduced a powerful method, known as the $Tangent\ Method$, 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.
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Cited literature [8 references]

https://hal-cea.archives-ouvertes.fr/cea-01692535
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Submitted on : Thursday, January 25, 2018 - 11:31:05 AM
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### Citation

Philippe Di Francesco, Matthew Lapa. Arctic Curves In Path Models from The Tangent Method. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2018, 51, pp.155202. ⟨10.1088/1751-8121/aab3c0⟩. ⟨cea-01692535⟩

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