等径图上的$Z$ -不变大质量Laplacian
The $Z$-invariant massive Laplacian on isoradial graphs
摘要：
我们引入了一个单参数族的大规模拉普拉斯算子$(Δ^{m(k)})_{k\in[0,1)}$定义在等径图，涉及椭圆函数。我们证明了一个显式公式的逆$Δ^{m(k)}$，大量的格林函数，它具有显着的性质，只依赖于局部几何的图，并计算其渐近。研究了随机根生成森林的相应统计力学模型。我们证明了一个明确的本地公式的无限体积玻尔兹曼措施，和模型的自由能。我们表明，该模型在$k=0$经历了二阶相变，从而证明了生成树对应于由凯尼恩介绍的拉普拉斯算子是关键的。证明了大规模Laplacian算子$(Δ^{m(k)})_{k\in(0,1)}$提供了一个单参数的$Z$ -不变根生成森林模型族.当等径图还是$\mathbb{Z}^2$ -周期图时，我们考虑了大质量拉普拉斯算子的特征多项式的谱曲线.我们提供了一个明确的参数化曲线，并证明它是Harnack和有亏格$1$。我们进一步证明了具有$(z,w)\leftrightarrow(z^{-1},w^{-1})$对称性的$1$亏格的每一条Harnack曲线都来自于这样一个大质量的Laplacian。
英文摘要：
We introduce a one-parameter family of massive Laplacian operators $(Δ^{m(k)})_{k\in[0,1)}$ defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of $Δ^{m(k)}$, the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at $k=0$, thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon are critical. We prove that the massive Laplacian operators $(Δ^{m(k)})_{k\in(0,1)}$ provide a one-parameter family of $Z$-invariant rooted spanning forest models. When the isoradial graph is moreover $\mathbb{Z}^2$-periodic, we consider the spectral curve of the characteristic polynomial of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus $1$. We further show that every Harnack curve of genus $1$ with $(z,w)\leftrightarrow(z^{-1},w^{-1})$ symmetry arises from such a massive Laplacian.
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