北极圈再访
The Arctic Circle Revisited
摘要：
通过考虑一个特别定制的体关联函数--空洞形成概率，解决了具有畴壁边界条件的六顶点模型中的极限形状问题。给出了该关联函数的闭表达式，包括行列式和多重积分，从而可以对顶点权值全范围内的模型极限形状进行系统的处理。具体来说，我们表明，顶点权重对应于相图上的自由费米子线，空形成概率与一个矩阵模型与三重对数奇异性，或三重Penner模型。这个模型的鞍点分析导致了北极圈定理，以及它对北极椭圆的推广，这是以前从多米诺骨牌中得知的。
英文摘要：
The problem of limit shapes in the six-vertex model with domain wall boundary conditions is addressed by considering a specially tailored bulk correlation function, the emptiness formation probability. A closed expression of this correlation function is given, both in terms of certain determinant and multiple integral, which allows for a systematic treatment of the limit shapes of the model for full range of values of vertex weights. Specifically, we show that for vertex weights corresponding to the free-fermion line on the phase diagram, the emptiness formation probability is related to a one-matrix model with a triple logarithmic singularity, or Triple Penner model. The saddle-point analysis of this model leads to the Arctic Circle Theorem, and its generalization to the Arctic Ellipses, known previously from domino tilings.
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