极小图上的椭圆二聚体与亏格为1的Harnack曲线
Elliptic dimers on minimal graphs and genus 1 Harnack curves
摘要：
本文对具有Fock椭圆权的无限极小图上的二聚体模型[arXiv：1503.00289]进行了全面的研究。在[arXiv：052711，arXiv：1612.09082，arXiv 1801.00207]中研究了这种模型的具体实例;我们现在处理一般亏格1的情况，从而证明了[arXiv：math-ph/0202018，arXiv：math/0311062]关于等径临界模型的亏格0结果的非平凡扩展。我们给出了Kasteleyn算子的双参数逆族的显式局部表达式，其中基础图上没有周期性假设。当最小图满足一个自然条件时，我们从这些逆中构造了一个二聚体Gibbs测度族，并通过推导每个相位中相关性的渐近性来描述模型的相图。在$\mathbb{Z}^2$周期的情况下，这给出了另一种描述的遍历吉布斯措施在[arXiv：math-ph/0311005]中构建的凯尼恩，Okounkov和谢菲尔德。我们还建立了周期极小图上的椭圆二聚体模型与亏格为1的Harnack曲线之间的对应关系。最后，我们证明了一个二分二聚体模型是不变的收缩/膨胀下的2价顶点和蜘蛛移动当且仅当相关的Kasteleyn系数是反对称的，并满足Fay的三割线身份。
英文摘要：
This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock's elliptic weights [arXiv:1503.00289]. Specific instances of such models were studied in [arXiv:052711, arXiv:1612.09082, arXiv1801.00207]; we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of [arXiv:math-ph/0202018, arXiv:math/0311062] on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the $\mathbb{Z}^2$-periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in [arXiv:math-ph/0311005] by Kenyon, Okounkov and Sheffield. We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay's trisecant identity.
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