In this paper we consider the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo and Sportiello (J Stat Phys 164:1488–1523, 2016) as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.

中文翻译：

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三束域冰模型的北极边界

在本文中，我们考虑任意三束域上冰点的六顶点模型，它是正方形（或等效地，均匀随机交替符号矩阵）上的域壁冰模型的推广。我们表明该模型表现出北极边界现象，其边界由显式代数曲线的并集给出。这最初是由 Colomo 和 Sportiello (J Stat Phys 164:1488–1523, 2016) 预测的，作为他们引入的用于定位北极边界的一般启发式方法的初始应用之一，称为（几何）切线方法。我们的证明使用非交叉有向路径集合的概率分析来提供在这种情况下它们的切线方法启发式的数学证明，这可能是独立的兴趣。