Arctic Curves In Path Models from The Tangent Method
International audience 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 vario...
Published in: | Journal of Physics A: Mathematical and Theoretical |
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Main Authors: | , |
Other Authors: | , , , , , |
Format: | Article in Journal/Newspaper |
Language: | English |
Published: |
HAL CCSD
2018
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Subjects: | |
Online Access: | https://cea.hal.science/cea-01692535 https://cea.hal.science/cea-01692535/document https://cea.hal.science/cea-01692535/file/1711.03182.pdf https://doi.org/10.1088/1751-8121/aab3c0 |
Summary: | International audience 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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