Arctic curves in path models from the tangent method
Arctic curves in path models from the tangent method
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切线法路径模型中的北极曲线
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Matthew F. Lapa
中科院分区：
文献类型：
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作者：
P. Di Francesco;Matthew F. Lapa
最近，Colomo和Sportiello介绍了一种强大的方法，称为切线方法，用于计算统计模型中的北极曲线，该模型具有（非或弱）相交的晶格路径公式。我们应用切线方法计算各种模型中的北极曲线：阿兹特克钻石的多米诺瓷砖，我们恢复了著名的北极圈;一个模型的戴克路径相当于菱形瓷砖的半六边形，我们发现一个北极半椭圆;另一个菱形瓷砖模型与北极抛物线;垂直对称交替符号矩阵，我们发现相同的北极曲线为无约束交替符号矩阵。后一种情况涉及不相交但允许有密切接触点的晶格路径，切线方法仍然适用。对于每个问题，我们估计的大尺寸渐近的某个一点函数使用LU分解相应的Gessel-Viennot矩阵，并重新制定的结果服从渐近分析。
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.