We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

中文翻译：

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反铁磁 Ising 模型的 Lee-Yang 零点

我们研究了反铁磁伊辛模型的配分函数的零点位置，重点关注位于单位圆上的零点。我们给出了有根凯莱树类的精确表征，表明在最有趣的圆弧上零点并不密集。相比之下，我们证明，当考虑具有给定度数界限的所有图时，零点在圆形子弧中是密集的，这意味着凯莱树在这个意义上不是极值的。证明依赖于描述在考虑递归定义的树上的分区函数比率时出现的理性动态系统。