论文标题
最小图和属1 Harnack曲线的椭圆二聚体
Elliptic dimers on minimal graphs and genus 1 Harnack curves
论文作者
论文摘要
本文提供了有关Fock的椭圆重量的无限最小图的二聚体模型的全面研究[Arxiv：1503.00289]。在[arxiv：052711，arxiv：1612.09082，arxiv1801.00207]中研究了此类模型的具体实例；现在，我们处理一般属1病例，因此证明了[Arxiv：Math-PH/0202018，ARXIV：MATH/0311062]的属0结果的非平地扩展。我们为Kasteleyn运算符的两参数媒介家族提供了明确的局部表达，而在基础图上没有周期性假设。当最小图满足自然条件时，我们构建了一个二聚体吉布斯家族的测量，并通过在每个阶段中得出相关性的渐近学来描述模型的相图。在$ \ mathbb {z}^2 $ - 周期案例中，这提供了对[Arxiv：Math-ph/0311005]构建的完整的ergodic Gibbs测量的替代描述。我们还建立了椭圆形二聚体模型在周期性最小图与属1的竖琴曲线上之间的对应关系。最后，我们表明，在且仅当相关的kasteleyn系数抗敌对和满足的情况下，且仅当且仅当且仅当相关的Kasteleyn系数在2个价值的顶点的缩小/扩展下是不变的，并且蜘蛛在缩小/扩展下移动。
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.
