平面算法与八顶点模型的完整复杂性分类
New Planar Algorithms and a Full Complexity Classification of the Eight-Vertex Model
AI总结：
本文提出了一种平面八顶点模型的完整复杂性分类，揭示了新的可计算模型及与双色Ising模型等的非局部联系。
摘要：
我们为平面八顶点模型建立了完整的复杂性分类定理。对于八顶点模型中的每一个参数设置，在复数域中，配分函数要么（1）对于每个图都能在P时间内计算，要么（2）对于一般图是#P难的，但对平面图可在P时间内计算，要么（3）即使对于平面图也是#P难的。该分类有显式的判据。在（2）中，我们发现了新的可在平面图上计算的八顶点模型，这些模型通过一种组合变换得到，该变换将问题转换为平面Even Coloring问题，然后通过一种全息变换转换到平面六顶点模型中的可计算情况。在过程中，我们还发现了平面八顶点模型与双色Ising模型、共形晶格插值以及来自复分析的莫比乌斯变换之间的非局部联系。证明还利用了循环域。
英文摘要：
We prove a complete complexity classification theorem for the planar eight-vertex model. For every parameter setting in ${\mathbb C}$ for the eight-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) \#P-hard for general graphs but computable in P-time for planar graphs, or (3) \#P-hard even for planar graphs. The classification has an explicit criterion. In (2), we discover new P-time computable eight-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. They are obtained by a combinatorial transformation to the planar {\sc Even Coloring} problem followed by a holographic transformation to the tractable cases in the planar six-vertex model. In the process, we also encounter non-local connections between the planar eight vertex model and the bipartite Ising model, conformal lattice interpolation and Möbius transformation from complex analysis. The proof also makes use of cyclotomic fields.
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